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2. 51 find the distance between parallel planes and student project distance between two skew lines figure 2. 73 industrial pipe installations often feature pipes running in different directions. how can we find the distance between two skew pipes? finding the distance from a point to a line or from a line to a plane seems like a pretty abstract procedure. but, if the lines represent pipes in a chemical plant or tubes in an oil refinery or roads at an intersection of highways, confirming that the distance between them meets specifications can be both important and awkward to measure. one way is to model the two pipes as lines, using the techniques in this chapter, and then calculate the distance between them. the calculation involves forming vectors along the directions of the lines and using both the cross product and the dot product. 184 2 • vectors in space access for free at openstax. org the symmetric forms of two lines, and are you are to develop a formula for the distance between these two lines, in terms of the values the distance between two lines is usually taken to mean the minimum distance, so this is the length of a line segment or the length of a vector that is perpendicular to both lines and intersects both lines. 1. first, write down two vectors, and that lie along and respectively. 2. find the cross product of these two vectors and call it this vector is perpendicular to and hence is perpendicular to both lines. 3. from vector form a unit vector in the same direction. 4. use symmetric equations to find a convenient vector that lies between any two points, one on each line. again, this can be done directly from the symmetric equations. 5. the dot product of two vectors is the magnitude of the projection of one vector onto the other times the magnitude of the other vector — that is, where is the angle between the vectors. using the dot product, find the projection of vector found in step onto unit vector found in step 3. this projection is perpendicular to both lines, and hence its length must be the perpendicular distance between them. note that the value of may be negative, depending on your choice of vector or the order of the cross product, so use absolute value signs around the numerator. 6. check that your formula | openstax_calculus_volume_3_-_web | [
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this projection is perpendicular to both lines, and hence its length must be the perpendicular distance between them. note that the value of may be negative, depending on your choice of vector or the order of the cross product, so use absolute value signs around the numerator. 6. check that your formula gives the correct distance of between the following two lines : 7. is your general expression valid when the lines are parallel? if not, why not? ( hint : what do you know about the value of the cross product of two parallel vectors? where would that result show up in your expression for 8. demonstrate that your expression for the distance is zero when the lines intersect. recall that two lines intersect if they are not parallel and they are in the same plane. hence, consider the direction of and what is the result of their dot product? 9. consider the following application. engineers at a refinery have determined they need to install support struts between many of the gas pipes to reduce damaging vibrations. to minimize cost, they plan to install these struts at the closest points between adjacent skewed pipes. because they have detailed schematics of the structure, they are able to determine the correct lengths of the struts needed, and hence manufacture and distribute them to the installation crews without spending valuable time making measurements. the rectangular frame structure has the dimensions ( height, width, and depth ). one sector has a pipe entering the lower corner of the standard frame unit and exiting at the diametrically opposed corner ( the one farthest away at the top ) ; call this a second pipe enters and exits at the two different opposite lower corners ; call this ( figure 2. 74 ). | openstax_calculus_volume_3_-_web | [
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2. 5 • equations of lines and planes in space 185 figure 2. 74 two pipes cross through a standard frame unit. write down the vectors along the lines representing those pipes, find the cross product between them from which to create the unit vector define a vector that spans two points on each line, and finally determine the minimum distance between the lines. ( take the origin to be at the lower corner of the first pipe. ) similarly, you may also develop the symmetric equations for each line and substitute directly into your formula. section 2. 5 exercises in the following exercises, points and are given. let be the line passing through points and a. find the vector equation of line b. find parametric equations of line c. find symmetric equations of line d. find parametric equations of the line segment determined by and 243. 244. 245. 246. for the following exercises, point and vector are given. let be the line passing through point with direction a. find parametric equations of line b. find symmetric equations of line c. find the intersection of the line with the xy - plane. 247. 248. 249. where and 250. where and for the following exercises, line is given. a. find point that belongs to the line and direction vector of the line. express in component form. b. find the distance from the origin to line 186 2 • vectors in space access for free at openstax. org 251. ℝ 252. 253. find the distance between point and the line of symmetric equations 254. find the distance between point and the line of parametric equations ℝ for the following exercises, lines and are given. a. verify whether lines and are parallel. b. if the lines and are parallel, then find the distance between them. 255. ℝ 256. ℝ 257. show that the line passing through points and is perpendicular to the line with equation ℝ 258. are the lines of equations and ℝperpendicular to each other? 259. find the point of intersection of the lines of equations and ℝ 260. find the intersection point of the x - axis with the line of parametric equations ℝ for the following exercises, lines and are given. determine whether the lines are equal, parallel but | openstax_calculus_volume_3_-_web | [
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##perpendicular to each other? 259. find the point of intersection of the lines of equations and ℝ 260. find the intersection point of the x - axis with the line of parametric equations ℝ for the following exercises, lines and are given. determine whether the lines are equal, parallel but not equal, skew, or intersecting. 261. and 262. ℝ and ℝ 263. ℝand 264. and ℝ | openstax_calculus_volume_3_-_web | [
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2. 5 • equations of lines and planes in space 187 265. consider line of symmetric equations and point a. find parametric equations for a line parallel to that passes through point b. find symmetric equations of a line skew to and that passes through point c. find symmetric equations of a line that intersects and passes through point 266. consider line of parametric equations ℝ a. find parametric equations for a line parallel to that passes through the origin. b. find parametric equations of a line skew to that passes through the origin. c. find symmetric equations of a line that intersects and passes through the origin. for the following exercises, point and vector are given. a. find the scalar equation of the plane that passes through and has normal vector b. find the general form of the equation of the plane that passes through and has normal vector 267. 268. 269. 270. for the following exercises, the equation of a plane is given. a. find normal vector to the plane. express using standard unit vectors. b. find the intersections of the plane with the coordinate axes. c. sketch the plane. 271. [ t ] 272. 273. 274. 275. given point and vector find point on the x - axis such that and are orthogonal. 276. show there is no plane perpendicular to that passes through points and 277. find parametric equations of the line passing through point that is perpendicular to the plane of equation 278. find symmetric equations of the line passing through point that is perpendicular to the plane of equation 279. show that line is parallel to plane 280. find the real number such that the line of parametric equations ℝis parallel to the plane of equation 188 2 • vectors in space access for free at openstax. org for the following exercises, points are given. a. find the general equation of the plane passing through b. write the vector equation of the plane at a., where is an arbitrary point of the plane. c. find parametric equations of the line passing through the origin that is perpendicular to the plane passing through 281. and 282. and 283. consider the planes of equations and a. show that the planes intersect. b. find symmetric equations of the line | openstax_calculus_volume_3_-_web | [
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where is an arbitrary point of the plane. c. find parametric equations of the line passing through the origin that is perpendicular to the plane passing through 281. and 282. and 283. consider the planes of equations and a. show that the planes intersect. b. find symmetric equations of the line passing through point that is parallel to the line of intersection of the planes. 284. consider the planes of equations and a. show that the planes intersect. b. find parametric equations of the line passing through point that is parallel to the line of intersection of the planes. 285. find the scalar equation of the plane that passes through point and is perpendicular to the line of intersection of planes and 286. find the general equation of the plane that passes through the origin and is perpendicular to the line of intersection of planes and 287. determine whether the line of parametric equations ℝintersects the plane with equation if it does intersect, find the point of intersection. 288. determine whether the line of parametric equations ℝintersects the plane with equation if it does intersect, find the point of intersection. 289. find the distance from point to the plane of equation 290. find the distance from point to the plane of equation for the following exercises, the equations of two planes are given. a. determine whether the planes are parallel, orthogonal, or neither. b. if the planes are neither parallel nor orthogonal, then find the measure of the angle between the planes. express the answer in degrees rounded to the nearest integer. 291. [ t ] 292. 293. | openstax_calculus_volume_3_-_web | [
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2. 5 • equations of lines and planes in space 189 294. [ t ] 295. show that the lines of equations ℝand are skew, and find the distance between them. 296. show that the lines of equations ℝand ℝare skew, and find the distance between them. 297. consider point and the plane of equation a. find the radius of the sphere with center tangent to the given plane. b. find point p of tangency. 298. consider the plane of equation a. find the equation of the sphere with center at the origin that is tangent to the given plane. b. find parametric equations of the line passing through the origin and the point of tangency. 299. two children are playing with a ball. the girl throws the ball to the boy. the ball travels in the air, curves ft to the right, and falls ft away from the girl ( see the following figure ). if the plane that contains the trajectory of the ball is perpendicular to the ground, find its equation. 190 2 • vectors in space access for free at openstax. org 300. [ t ] john allocates dollars to consume monthly three goods of prices in this context, the budget equation is defined as where and represent the number of items bought from each of the goods. the budget set is given by and the budget plane is the part of the plane of equation for which and consider and a. use a cas to graph the budget set and budget plane. b. for find the new budget equation and graph the budget set in the same system of coordinates. 301. [ t ] consider the position vector of a particle at time where the components of r are expressed in centimeters and time is measured in seconds. let be the position vector of the particle after sec. a. determine the velocity vector of the particle after sec. b. find the scalar equation of the plane that is perpendicular to and passes through point this plane is called the normal plane to the path of the particle at point c. use a cas to visualize the path of the particle along with the velocity vector and normal plane at point | openstax_calculus_volume_3_-_web | [
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2. 5 • equations of lines and planes in space 191 302. [ t ] a solar panel is mounted on the roof of a house. the panel may be regarded as positioned at the points of coordinates ( in meters ) and ( see the following figure ). a. find the general form of the equation of the plane that contains the solar panel by using points and show that its normal vector is equivalent to b. find parametric equations of line that passes through the center of the solar panel and has direction vector which points toward the position of the sun at a particular time of day. c. find symmetric equations of line that passes through the center of the solar panel and is perpendicular to it. d. determine the angle of elevation of the sun above the solar panel by using the angle between lines and | openstax_calculus_volume_3_-_web | [
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2. 6 quadric surfaces learning objectives 2. 6. 1 identify a cylinder as a type of three - dimensional surface. 2. 6. 2 recognize the main features of ellipsoids, paraboloids, and hyperboloids. 2. 6. 3 use traces to draw the intersections of quadric surfaces with the coordinate planes. we have been exploring vectors and vector operations in three - dimensional space, and we have developed equations to describe lines, planes, and spheres. in this section, we use our knowledge of planes and spheres, which are examples of three - dimensional figures called surfaces, to explore a variety of other surfaces that can be graphed in a three - dimensional coordinate system. identifying cylinders the first surface we ’ ll examine is the cylinder. although most people immediately think of a hollow pipe or a soda straw when they hear the word cylinder, here we use the broad mathematical meaning of the term. as we have seen, cylindrical surfaces don ’ t have to be circular. a rectangular heating duct is a cylinder, as is a rolled - up yoga mat, the cross - section of which is a spiral shape. in the two - dimensional coordinate plane, the equation describes a circle centered at the origin with radius in three - dimensional space, this same equation represents a surface. imagine copies of a circle stacked on top of each other centered on the z - axis ( figure 2. 75 ), forming a hollow tube. we can then construct a cylinder from the set of lines parallel to the z - axis passing through circle in the xy - plane, as shown in the figure. in this way, any curve in 192 2 • vectors in space access for free at openstax. org one of the coordinate planes can be extended to become a surface. figure 2. 75 in three - dimensional space, the graph of equation is a cylinder with radius centered on the z - axis. it continues indefinitely in the positive and negative directions. definition a set of lines parallel to a given line passing through a given curve is known as a cylindrical surface, or cylinder. the parallel lines are called rulings. from this definition, we can see that we still have a cylinder in three - dimensional space, even if the curve is | openstax_calculus_volume_3_-_web | [
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negative directions. definition a set of lines parallel to a given line passing through a given curve is known as a cylindrical surface, or cylinder. the parallel lines are called rulings. from this definition, we can see that we still have a cylinder in three - dimensional space, even if the curve is not a circle. any curve can form a cylinder, and the rulings that compose the cylinder may be parallel to any given line ( figure 2. 76 ). figure 2. 76 in three - dimensional space, the graph of equation is a cylinder, or a cylindrical surface with rulings parallel to the y - axis. example 2. 55 graphing cylindrical surfaces sketch the graphs of the following cylindrical surfaces. a. | openstax_calculus_volume_3_-_web | [
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2. 6 • quadric surfaces 193 b. c. solution a. the variable can take on any value without limit. therefore, the lines ruling this surface are parallel to the y - axis. the intersection of this surface with the xz - plane forms a circle centered at the origin with radius ( see the following figure ). figure 2. 77 the graph of equation is a cylinder with radius centered on the y - axis. b. in this case, the equation contains all three variables and so none of the variables can vary arbitrarily. the easiest way to visualize this surface is to use a computer graphing utility ( see the following figure ). figure 2. 78 c. in this equation, the variable z can take on any value without limit. therefore, the lines composing this surface are parallel to the z - axis. the intersection of this surface with the xy - plane outlines curve ( see the following figure ). 194 2 • vectors in space access for free at openstax. org figure 2. 79 the graph of equation is formed by a set of lines parallel to the z - axis passing through curve in the xy - plane. 2. 52 sketch or use a graphing tool to view the graph of the cylindrical surface defined by equation when sketching surfaces, we have seen that it is useful to sketch the intersection of the surface with a plane parallel to one of the coordinate planes. these curves are called traces. we can see them in the plot of the cylinder in figure 2. 80. definition the traces of a surface are the cross - sections created when the surface intersects a plane parallel to one of the coordinate planes. | openstax_calculus_volume_3_-_web | [
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2. 6 • quadric surfaces 195 figure 2. 80 ( a ) this is one view of the graph of equation ( b ) to find the trace of the graph in the xz - plane, set the trace is simply a two - dimensional sine wave. traces are useful in sketching cylindrical surfaces. for a cylinder in three dimensions, though, only one set of traces is useful. notice, in figure 2. 80, that the trace of the graph of in the xz - plane is useful in constructing the graph. the trace in the xy - plane, though, is just a series of parallel lines, and the trace in the yz - plane is simply one line. cylindrical surfaces are formed by a set of parallel lines. not all surfaces in three dimensions are constructed so simply, however. we now explore more complex surfaces, and traces are an important tool in this investigation. quadric surfaces we have learned about surfaces in three dimensions described by first - order equations ; these are planes. some other common types of surfaces can be described by second - order equations. we can view these surfaces as three - dimensional extensions of the conic sections we discussed earlier : the ellipse, the parabola, and the hyperbola. we call these graphs quadric surfaces. definition quadric surfaces are the graphs of equations that can be expressed in the form when a quadric surface intersects a coordinate plane, the trace is a conic section. an ellipsoid is a surface described by an equation of the form set to see the trace of the ellipsoid in the yz - plane. to see the traces in the xy - and xz - planes, set and respectively. notice that, if the trace in the xy - plane is a circle. similarly, if the trace in the xz - plane is a circle and, if then the trace in the yz - plane is a circle. a sphere, then, is an ellipsoid with example 2. 56 sketching an ellipsoid sketch the ellipsoid 196 2 • vectors in space access for free at openstax. org solution start by sketching the traces. to find the trace in the | openstax_calculus_volume_3_-_web | [
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circle. a sphere, then, is an ellipsoid with example 2. 56 sketching an ellipsoid sketch the ellipsoid 196 2 • vectors in space access for free at openstax. org solution start by sketching the traces. to find the trace in the xy - plane, set ( see figure 2. 81 ). to find the other traces, first set and then set figure 2. 81 ( a ) this graph represents the trace of equation in the xy - plane, when we set ( b ) when we set we get the trace of the ellipsoid in the xz - plane, which is an ellipse. ( c ) when we set we get the trace of the ellipsoid in the yz - plane, which is also an ellipse. now that we know what traces of this solid look like, we can sketch the surface in three dimensions ( figure 2. 82 ). figure 2. 82 ( a ) the traces provide a framework for the surface. ( b ) the center of this ellipsoid is the origin. the trace of an ellipsoid is an ellipse in each of the coordinate planes. however, this does not have to be the case for all quadric surfaces. many quadric surfaces have traces that are different kinds of conic sections, and this is usually indicated by the name of the surface. for example, if a surface can be described by an equation of the form then we call that surface an elliptic paraboloid. the trace in the xy - plane is an ellipse, but the traces in the xz - plane and yz - plane are parabolas ( figure 2. 83 ). other elliptic paraboloids can have other orientations simply by | openstax_calculus_volume_3_-_web | [
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2. 6 • quadric surfaces 197 interchanging the variables to give us a different variable in the linear term of the equation or figure 2. 83 this quadric surface is called an elliptic paraboloid. example 2. 57 identifying traces of quadric surfaces describe the traces of the elliptic paraboloid solution to find the trace in the xy - plane, set the trace in the plane is simply one point, the origin. since a single point does not tell us what the shape is, we can move up the z - axis to an arbitrary plane to find the shape of other traces of the figure. the trace in plane is the graph of equation which is an ellipse. in the xz - plane, the equation becomes the trace is a parabola in this plane and in any plane with the equation in planes parallel to the yz - plane, the traces are also parabolas, as we can see in the following figure. 198 2 • vectors in space access for free at openstax. org figure 2. 84 ( a ) the paraboloid ( b ) the trace in plane ( c ) the trace in the xz - plane. ( d ) the trace in the yz - plane. 2. 53 a hyperboloid of one sheet is any surface that can be described with an equation of the form describe the traces of the hyperboloid of one sheet given by equation hyperboloids of one sheet have some fascinating properties. for example, they can be constructed using straight lines, such as in the sculpture in figure 2. 85 ( a ). in fact, cooling towers for nuclear power plants are often constructed in the shape of a hyperboloid. the builders are able to use straight steel beams in the construction, which makes the towers very strong while using relatively little material ( figure 2. 85 ( b ) ). | openstax_calculus_volume_3_-_web | [
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2. 6 • quadric surfaces 199 figure 2. 85 ( a ) a sculpture in the shape of a hyperboloid can be constructed of straight lines. ( b ) cooling towers for nuclear power plants are often built in the shape of a hyperboloid. example 2. 58 chapter opener : finding the focus of a parabolic reflector energy hitting the surface of a parabolic reflector is concentrated at the focal point of the reflector ( figure 2. 86 ). if the surface of a parabolic reflector is described by equation where is the focal point of the reflector? figure 2. 86 energy reflects off of the parabolic reflector and is collected at the focal point. ( credit : modification of cgp grey, wikimedia commons ) solution since z is the first - power variable, the axis of the reflector corresponds to the z - axis. the coefficients of and are equal, so the cross - section of the paraboloid perpendicular to the z - axis is a circle. we can consider a trace in the xz - plane or the yz - plane ; the result is the same. setting the trace is a parabola opening up along the z - axis, with standard equation where is the focal length of the parabola. in this case, this equation becomes or so p is m, which tells us that the focus of the paraboloid is m up the axis from the vertex. because the vertex of this surface is the origin, the focal point is seventeen standard quadric surfaces can be derived from the general equation the following figures summarizes the most important ones. in the following two figures, the “ axis ” of a quadric surface may or may not be an axis of symmetry. however, all traces on the surface formed by any plane perpendicular to an “ axis ” will be of the same conic section type. 200 2 • vectors in space access for free at openstax. org figure 2. 87 characteristics of common quadratic surfaces : ellipsoid, hyperboloid of one sheet, hyperboloid of two sheets. | openstax_calculus_volume_3_-_web | [
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2. 6 • quadric surfaces 201 figure 2. 88 characteristics of common quadratic surfaces : elliptic cone, elliptic paraboloid, hyperbolic paraboloid. example 2. 59 identifying equations of quadric surfaces identify the surfaces represented by the given equations. a. b. 202 2 • vectors in space access for free at openstax. org solution a. the and terms are all squared, and are all positive, so this is probably an ellipsoid. however, let ’ s put the equation into the standard form for an ellipsoid just to be sure. we have dividing through by 144 gives so, this is, in fact, an ellipsoid, centered at the origin. b. we first notice that the term is raised only to the first power, so this is either an elliptic paraboloid or a hyperbolic paraboloid. we also note there are terms and terms that are not squared, so this quadric surface is not centered at the origin. we need to complete the square to put this equation in one of the standard forms. we have this is an elliptic paraboloid centered at 2. 54 identify the surface represented by equation section 2. 6 exercises for the following exercises, sketch and describe the cylindrical surface of the given equation. 303. [ t ] 304. [ t ] 305. [ t ] 306. [ t ] 307. [ t ] 308. [ t ] for the following exercises, the graph of a quadric surface is given. a. specify the name of the quadric surface. b. determine the axis of the quadric surface. | openstax_calculus_volume_3_-_web | [
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2. 6 • quadric surfaces 203 309. 310. 311. 312. for the following exercises, match the given quadric surface with its corresponding equation in standard form. a. b. c. d. e. f. 313. hyperboloid of two sheets 314. ellipsoid 315. elliptic paraboloid 316. hyperbolic paraboloid 317. hyperboloid of one sheet 318. elliptic cone for the following exercises, rewrite the given equation of the quadric surface in standard form. identify the surface. 319. 320. 321. 322. 323. 324. 325. 326. 327. 328. 329. 330. for the following exercises, find the trace of the given quadric surface in the specified plane of coordinates and sketch it. 331. [ t ] 332. [ t ] 333. [ t ] 204 2 • vectors in space access for free at openstax. org 334. [ t ] 335. [ t ] 336. [ t ] 337. use the graph of the given quadric surface to answer the questions. a. specify the name of the quadric surface. b. which of the equations — or — corresponds to the graph? c. use b. to write the equation of the quadric surface in standard form. 338. use the graph of the given quadric surface to answer the questions. a. specify the name of the quadric surface. b. which of the equations — — corresponds to the graph above? c. use b. to write the equation of the quadric surface in standard form. for the following exercises, the equation of a quadric surface is given. a. use the method of completing the square to write the equation in standard form. b. identify the surface. 339. 340. 341. 342. | openstax_calculus_volume_3_-_web | [
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2. 6 • quadric surfaces 205 343. 344. 345. write the standard form of the equation of the ellipsoid centered at the origin that passes through points and 346. write the standard form of the equation of the ellipsoid centered at point that passes through points and 347. determine the intersection points of elliptic cone with the line of symmetric equations 348. determine the intersection points of parabolic hyperboloid with the line of parametric equations where ℝ 349. find the equation of the quadric surface with points that are equidistant from point and plane of equation identify the surface. 350. find the equation of the quadric surface with points that are equidistant from point and plane of equation identify the surface. 351. if the surface of a parabolic reflector is described by equation find the focal point of the reflector. 352. consider the parabolic reflector described by equation find its focal point. 353. show that quadric surface reduces to two parallel planes. 206 2 • vectors in space access for free at openstax. org 354. show that quadric surface reduces to two parallel planes passing. 355. [ t ] the intersection between cylinder and sphere is called a viviani curve. a. solve the system consisting of the equations of the surfaces to find the equations of the intersection curve. ( hint : find and in terms of b. use a computer algebra system ( cas ) to visualize the intersection curve on sphere 356. hyperboloid of one sheet and elliptic cone are represented in the following figure along with their intersection curves. identify the intersection curves and find their equations ( hint : find y from the system consisting of the equations of the surfaces. ) 357. [ t ] use a cas to create the intersection between cylinder and ellipsoid and find the equations of the intersection curves. 358. [ t ] a spheroid is an ellipsoid with two equal semiaxes. for instance, the equation of a spheroid with the z - axis as its axis of symmetry is given by where and are positive real numbers. the spheroid is called oblate if and prolate for a. the | openstax_calculus_volume_3_-_web | [
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##oid is an ellipsoid with two equal semiaxes. for instance, the equation of a spheroid with the z - axis as its axis of symmetry is given by where and are positive real numbers. the spheroid is called oblate if and prolate for a. the eye cornea is approximated as a prolate spheroid with an axis that is the eye, where write the equation of the spheroid that models the cornea and sketch the surface. b. give two examples of objects with prolate spheroid shapes. | openstax_calculus_volume_3_-_web | [
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2. 6 • quadric surfaces 207 359. [ t ] in cartography, earth is approximated by an oblate spheroid rather than a sphere. the radii at the equator and poles are approximately mi and mi, respectively. a. write the equation in standard form of the ellipsoid that represents the shape of earth. assume the center of earth is at the origin and that the trace formed by plane corresponds to the equator. b. sketch the graph. c. find the equation of the intersection curve of the surface with plane that is parallel to the xy - plane. the intersection curve is called a parallel. d. find the equation of the intersection curve of the surface with plane that passes through the z - axis. the intersection curve is called a meridian. 360. [ t ] a set of buzzing stunt magnets ( or “ rattlesnake eggs ” ) includes two sparkling, polished, superstrong spheroid - shaped magnets well - known for children ’ s entertainment. each magnet is in. long and in. wide at the middle. while tossing them into the air, they create a buzzing sound as they attract each other. a. write the equation of the prolate spheroid centered at the origin that describes the shape of one of the magnets. b. write the equations of the prolate spheroids that model the shape of the buzzing stunt magnets. use a cas to create the graphs. 208 2 • vectors in space access for free at openstax. org 361. [ t ] a heart - shaped surface is given by equation a. use a cas to graph the surface that models this shape. b. determine and sketch the trace of the heart - shaped surface on the xz - plane. 362. [ t ] the ring torus symmetric about the z - axis is a special type of surface in topology and its equation is given by where the numbers and are called are the major and minor radii, respectively, of the surface. the following figure shows a ring torus for which a. write the equation of the ring torus with and use a cas to graph the surface. compare the graph with the figure given | openstax_calculus_volume_3_-_web | [
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given by where the numbers and are called are the major and minor radii, respectively, of the surface. the following figure shows a ring torus for which a. write the equation of the ring torus with and use a cas to graph the surface. compare the graph with the figure given. b. determine the equation and sketch the trace of the ring torus from a. on the xy - plane. c. give two examples of objects with ring torus shapes. | openstax_calculus_volume_3_-_web | [
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2. 7 cylindrical and spherical coordinates learning objectives 2. 7. 1 convert from cylindrical to rectangular coordinates. 2. 7. 2 convert from rectangular to cylindrical coordinates. 2. 7. 3 convert from spherical to rectangular coordinates. 2. 7. 4 convert from rectangular to spherical coordinates. the cartesian coordinate system provides a straightforward way to describe the location of points in space. some surfaces, however, can be difficult to model with equations based on the cartesian system. this is a familiar problem ; recall that in two dimensions, polar coordinates often provide a useful alternative system for describing the location of a point in the plane, particularly in cases involving circles. in this section, we look at two different ways of describing the location of points in space, both of them based on extensions of polar coordinates. as the name suggests, cylindrical coordinates are useful for dealing with problems involving cylinders, such as calculating the volume of a round water tank or the amount of oil flowing through a pipe. similarly, spherical coordinates are useful for dealing with problems involving spheres, such as finding the volume of domed structures. cylindrical coordinates when we expanded the traditional cartesian coordinate system from two dimensions to three, we simply added a new axis to model the third dimension. starting with polar coordinates, we can follow this same process to create a new three - dimensional coordinate system, called the cylindrical coordinate system. in this way, cylindrical coordinates provide a natural extension of polar coordinates to three dimensions. | openstax_calculus_volume_3_-_web | [
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2. 7 • cylindrical and spherical coordinates 209 definition in the cylindrical coordinate system, a point in space ( figure 2. 89 ) is represented by the ordered triple where • are the polar coordinates of the point ’ s projection in the xy - plane • is the usual in the cartesian coordinate system figure 2. 89 the right triangle lies in the xy - plane. the length of the hypotenuse is and is the measure of the angle formed by the positive x - axis and the hypotenuse. the z - coordinate describes the location of the point above or below the xy - plane. in the xy - plane, the right triangle shown in figure 2. 89 provides the key to transformation between cylindrical and cartesian, or rectangular, coordinates. theorem 2. 15 conversion between cylindrical and cartesian coordinates the rectangular coordinates and the cylindrical coordinates of a point are related as follows : as when we discussed conversion from rectangular coordinates to polar coordinates in two dimensions, it should be noted that the equation has an infinite number of solutions. however, if we restrict to values between and then we can find a unique solution based on the quadrant of the xy - plane in which original point is located. note that if then the value of is either or depending on the value of notice that these equations are derived from properties of right triangles. to make this easy to see, consider point in the xy - plane with rectangular coordinates and with cylindrical coordinates as shown in the following figure. 210 2 • vectors in space access for free at openstax. org figure 2. 90 the pythagorean theorem provides equation right - triangle relationships tell us that and let ’ s consider the differences between rectangular and cylindrical coordinates by looking at the surfaces generated when each of the coordinates is held constant. if is a constant, then in rectangular coordinates, surfaces of the form or are all planes. planes of these forms are parallel to the yz - plane, the xz - plane, and the xy - plane, respectively. when we convert to cylindrical coordinates, the z - coordinate does not change. therefore, in cylindrical coordinates, surfaces of the form are planes parallel to the xy - plane. now, let ’ | openstax_calculus_volume_3_-_web | [
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the yz - plane, the xz - plane, and the xy - plane, respectively. when we convert to cylindrical coordinates, the z - coordinate does not change. therefore, in cylindrical coordinates, surfaces of the form are planes parallel to the xy - plane. now, let ’ s think about surfaces of the form the points on these surfaces are at a fixed distance from the z - axis. in other words, these surfaces are vertical circular cylinders. last, what about the points on a surface of the form are at a fixed angle from the x - axis, which gives us a half - plane that starts at the z - axis ( figure 2. 91 and figure 2. 92 ). figure 2. 91 in rectangular coordinates, ( a ) surfaces of the form are planes parallel to the yz - plane, ( b ) surfaces of the form are planes parallel to the xz - plane, and ( c ) surfaces of the form are planes parallel to the xy - plane. figure 2. 92 in cylindrical coordinates, ( a ) surfaces of the form are vertical cylinders of radius ( b ) surfaces of the form are half - planes at angle from the x - axis, and ( c ) surfaces of the form are planes parallel to the xy - plane. example 2. 60 converting from cylindrical to rectangular coordinates plot the point with cylindrical coordinates and express its location in rectangular coordinates. 2. 7 • cylindrical and spherical coordinates 211 solution conversion from cylindrical to rectangular coordinates requires a simple application of the equations listed in conversion between cylindrical and cartesian coordinates : the point with cylindrical coordinates has rectangular coordinates ( see the following figure ). figure 2. 93 the projection of the point in the xy - plane is 4 units from the origin. the line from the origin to the point ’ s projection forms an angle of with the positive x - axis. the point lies units below the xy - plane. | openstax_calculus_volume_3_-_web | [
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2. 55 point has cylindrical coordinates. plot and describe its location in space using rectangular, or cartesian, coordinates. if this process seems familiar, it is with good reason. this is exactly the same process that we followed in introduction to parametric equations and polar coordinates to convert from polar coordinates to two - dimensional rectangular coordinates. example 2. 61 converting from rectangular to cylindrical coordinates convert the rectangular coordinates to cylindrical coordinates. solution use the second set of equations from conversion between cylindrical and cartesian coordinates to translate from rectangular to cylindrical coordinates : we choose the positive square root, so now, we apply the formula to find in this case, is negative and is positive, which means we must select the value of between and in this case, the z - coordinates are the same in both rectangular and cylindrical coordinates : 212 2 • vectors in space access for free at openstax. org the point with rectangular coordinates has cylindrical coordinates approximately equal to | openstax_calculus_volume_3_-_web | [
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2. 56 convert point from cartesian coordinates to cylindrical coordinates. the use of cylindrical coordinates is common in fields such as physics. physicists studying electrical charges and the capacitors used to store these charges have discovered that these systems sometimes have a cylindrical symmetry. these systems have complicated modeling equations in the cartesian coordinate system, which make them difficult to describe and analyze. the equations can often be expressed in more simple terms using cylindrical coordinates. for example, the cylinder described by equation in the cartesian system can be represented by cylindrical equation example 2. 62 identifying surfaces in the cylindrical coordinate system describe the surfaces with the given cylindrical equations. a. b. c. solution a. when the angle is held constant while and are allowed to vary, the result is a half - plane ( see the following figure ). figure 2. 94 in polar coordinates, the equation describes the ray extending diagonally through the first quadrant. in three dimensions, this same equation describes a half - plane. b. substitute into equation to express the rectangular form of the equation : this equation describes a sphere centered at the origin with radius ( see the following figure ). 2. 7 • cylindrical and spherical coordinates 213 figure 2. 95 the sphere centered at the origin with radius can be described by the cylindrical equation c. to describe the surface defined by equation is it useful to examine traces parallel to the xy - plane. for example, the trace in plane is circle the trace in plane is circle and so on. each trace is a circle. as the value of increases, the radius of the circle also increases. the resulting surface is a cone ( see the following figure ). figure 2. 96 the traces in planes parallel to the xy - plane are circles. the radius of the circles increases as increases. | openstax_calculus_volume_3_-_web | [
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2. 57 describe the surface with cylindrical equation spherical coordinates in the cartesian coordinate system, the location of a point in space is described using an ordered triple in which each coordinate represents a distance. in the cylindrical coordinate system, location of a point in space is described using two distances and an angle measure in the spherical coordinate system, we again use an ordered triple to describe the location of a point in space. in this case, the triple describes one distance and two angles. spherical coordinates make it simple to describe a sphere, just as cylindrical coordinates make it easy to describe a cylinder. grid 214 2 • vectors in space access for free at openstax. org lines for spherical coordinates are based on angle measures, like those for polar coordinates. definition in the spherical coordinate system, a point in space ( figure 2. 97 ) is represented by the ordered triple where • ( the greek letter rho ) is the distance between and the origin • is the same angle used to describe the location in cylindrical coordinates ; • ( the greek letter phi ) is the angle formed by the positive z - axis and line segment where is the origin and figure 2. 97 the relationship among spherical, rectangular, and cylindrical coordinates. by convention, the origin is represented as in spherical coordinates. theorem 2. 16 converting among spherical, cylindrical, and rectangular coordinates rectangular coordinates and spherical coordinates of a point are related as follows : if a point has cylindrical coordinates then these equations define the relationship between cylindrical and spherical coordinates. | openstax_calculus_volume_3_-_web | [
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2. 7 • cylindrical and spherical coordinates 215 the formulas to convert from spherical coordinates to rectangular coordinates may seem complex, but they are straightforward applications of trigonometry. looking at figure 2. 98, it is easy to see that then, looking at the triangle in the xy - plane with as its hypotenuse, we have the derivation of the formula for is similar. figure 2. 96 also shows that and solving this last equation for and then substituting ( from the first equation ) yields also, note that, as before, we must be careful when using the formula to choose the correct value of figure 2. 98 the equations that convert from one system to another are derived from right - triangle relationships. as we did with cylindrical coordinates, let ’ s consider the surfaces that are generated when each of the coordinates is held constant. let be a constant, and consider surfaces of the form points on these surfaces are at a fixed distance from the origin and form a sphere. the coordinate in the spherical coordinate system is the same as in the cylindrical coordinate system, so surfaces of the form are half - planes, as before. last, consider surfaces of the form the points on these surfaces are at a fixed angle from the z - axis and form a half - cone ( figure 2. 99 ). figure 2. 99 in spherical coordinates, surfaces of the form are spheres of radius ( a ), surfaces of the form are 216 2 • vectors in space access for free at openstax. org half - planes at an angle from the x - axis ( b ), and surfaces of the form are half - cones at an angle from the z - axis ( c ). example 2. 63 converting from spherical coordinates plot the point with spherical coordinates and express its location in both rectangular and cylindrical coordinates. solution use the equations in converting among spherical, cylindrical, and rectangular coordinates to translate between spherical and cylindrical coordinates ( figure 2. 100 ) : figure 2. 100 the projection of the point in the xy - plane is units from the origin. the line from the origin to the point ’ s projection forms an angle of with the positive x - axis. the point lies units above the xy - plane. the point with | openstax_calculus_volume_3_-_web | [
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100 ) : figure 2. 100 the projection of the point in the xy - plane is units from the origin. the line from the origin to the point ’ s projection forms an angle of with the positive x - axis. the point lies units above the xy - plane. the point with spherical coordinates has rectangular coordinates finding the values in cylindrical coordinates is equally straightforward : thus, cylindrical coordinates for the point are 2. 7 • cylindrical and spherical coordinates 217 | openstax_calculus_volume_3_-_web | [
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2. 58 plot the point with spherical coordinates and describe its location in both rectangular and cylindrical coordinates. example 2. 64 converting from rectangular coordinates convert the rectangular coordinates to both spherical and cylindrical coordinates. solution start by converting from rectangular to spherical coordinates : because then the correct choice for is there are actually two ways to identify we can use the equation a more simple approach, however, is to use equation we know that and so and therefore the spherical coordinates of the point are to find the cylindrical coordinates for the point, we need only find the cylindrical coordinates for the point are example 2. 65 identifying surfaces in the spherical coordinate system describe the surfaces with the given spherical equations. a. b. c. d. solution a. the variable represents the measure of the same angle in both the cylindrical and spherical coordinate systems. points with coordinates lie on the plane that forms angle with the positive x - axis. because the surface described by equation is the half - plane shown in figure 2. 101. 218 2 • vectors in space access for free at openstax. org figure 2. 101 the surface described by equation is a half - plane. b. equation describes all points in the spherical coordinate system that lie on a line from the origin forming an angle measuring rad with the positive z - axis. these points form a half - cone ( figure 2. 102 ). because there is only one value for that is measured from the positive z - axis, we do not get the full cone ( with two pieces ). figure 2. 102 the equation describes a cone. to find the equation in rectangular coordinates, use equation 2. 7 • cylindrical and spherical coordinates 219 this is the equation of a cone centered on the z - axis. c. equation describes the set of all points units away from the origin — a sphere with radius ( figure 2. 103 ). figure 2. 103 equation describes a sphere with radius d. to identify this surface, convert the equation from spherical to rectangular coordinates, using equations and the equation describes a sphere centered at point with radius | openstax_calculus_volume_3_-_web | [
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the equation from spherical to rectangular coordinates, using equations and the equation describes a sphere centered at point with radius | openstax_calculus_volume_3_-_web | [
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2. 59 describe the surfaces defined by the following equations. a. b. c. 220 2 • vectors in space access for free at openstax. org spherical coordinates are useful in analyzing systems that have some degree of symmetry about a point, such as the volume of the space inside a domed stadium or wind speeds in a planet ’ s atmosphere. a sphere that has cartesian equation has the simple equation in spherical coordinates. in geography, latitude and longitude are used to describe locations on earth ’ s surface, as shown in figure 2. 104. although the shape of earth is not a perfect sphere, we use spherical coordinates to communicate the locations of points on earth. let ’ s assume earth has the shape of a sphere with radius mi. we express angle measures in degrees rather than radians because latitude and longitude are measured in degrees. figure 2. 104 in the latitude – longitude system, angles describe the location of a point on earth relative to the equator and the prime meridian. let the center of earth be the center of the sphere, with the ray from the center through the north pole representing the positive z - axis. the prime meridian represents the trace of the surface as it intersects the xz - plane. the equator is the trace of the sphere intersecting the xy - plane. example 2. 66 converting latitude and longitude to spherical coordinates the latitude of columbus, ohio, is n and the longitude is w, which means that columbus is north of the equator. imagine a ray from the center of earth through columbus and a ray from the center of earth through the equator directly south of columbus. the measure of the angle formed by the rays is in the same way, measuring from the prime meridian, columbus lies to the west. express the location of columbus in spherical coordinates. solution the radius of earth is mi, so the intersection of the prime meridian and the equator lies on the positive x - axis. movement to the west is then described with negative angle measures, which shows that because columbus lies north of the equator, it lies south of the north pole, so in spherical coordinates, columbus lies at point 2. 60 sydney, australia is at and express sydney ’ s location in spherical coordinates. cylindrical and spherical coordinates give us the | openstax_calculus_volume_3_-_web | [
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then described with negative angle measures, which shows that because columbus lies north of the equator, it lies south of the north pole, so in spherical coordinates, columbus lies at point 2. 60 sydney, australia is at and express sydney ’ s location in spherical coordinates. cylindrical and spherical coordinates give us the flexibility to select a coordinate system appropriate to the problem at hand. a thoughtful choice of coordinate system can make a problem much easier to solve, whereas a poor choice can lead to unnecessarily complex calculations. in the following example, we examine several different problems and discuss how to select the best coordinate system for each one. | openstax_calculus_volume_3_-_web | [
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2. 7 • cylindrical and spherical coordinates 221 example 2. 67 choosing the best coordinate system in each of the following situations, we determine which coordinate system is most appropriate and describe how we would orient the coordinate axes. there could be more than one right answer for how the axes should be oriented, but we select an orientation that makes sense in the context of the problem. note : there is not enough information to set up or solve these problems ; we simply select the coordinate system ( figure 2. 105 ). a. find the center of gravity of a bowling ball. b. determine the velocity of a submarine subjected to an ocean current. c. calculate the pressure in a conical water tank. d. find the volume of oil flowing through a pipeline. e. determine the amount of leather required to make a football. figure 2. 105 ( credit : ( a ) modification of work by scl hua, wikimedia, ( b ) modification of work by dvidshub, flickr, ( c ) modification of work by michael malak, wikimedia, ( d ) modification of work by sean mack, wikimedia, ( e ) modification of work by elvert barnes, flickr ) solution a. clearly, a bowling ball is a sphere, so spherical coordinates would probably work best here. the origin should be located at the physical center of the ball. there is no obvious choice for how the x -, y - and z - axes should be oriented. bowling balls normally have a weight block in the center. one possible choice is to align the z - axis with the axis of symmetry of the weight block. b. a submarine generally moves in a straight line. there is no rotational or spherical symmetry that applies in this situation, so rectangular coordinates are a good choice. the z - axis should probably point upward. the x - and y - axes could be aligned to point east and north, respectively. the origin should be some convenient physical location, such as the starting position of the submarine or the location of a particular port. c. a cone has several kinds of symmetry. in cylindrical coordinates, a cone can be represented by equation where is a constant. in spherical coordinates, we have seen | openstax_calculus_volume_3_-_web | [
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respectively. the origin should be some convenient physical location, such as the starting position of the submarine or the location of a particular port. c. a cone has several kinds of symmetry. in cylindrical coordinates, a cone can be represented by equation where is a constant. in spherical coordinates, we have seen that surfaces of the form are half - cones. last, in rectangular coordinates, elliptic cones are quadric surfaces and can be represented by equations of the form in this case, we could choose any of the three. however, the equation for the surface is more complicated in rectangular coordinates than in the other two systems, so we might want to avoid that choice. in addition, we are talking about a water tank, and the depth of the water might come into play at some point in our calculations, so it might be nice to have a component that represents height and depth directly. based on this reasoning, cylindrical coordinates might be the best choice. choose the z - axis to align with the axis of the cone. the orientation of the other two axes is arbitrary. the origin should be the bottom point of the cone. d. a pipeline is a cylinder, so cylindrical coordinates would be best the best choice. in this case, however, we would 222 2 • vectors in space access for free at openstax. org likely choose to orient our z - axis with the center axis of the pipeline. the x - axis could be chosen to point straight downward or to some other logical direction. the origin should be chosen based on the problem statement. note that this puts the z - axis in a horizontal orientation, which is a little different from what we usually do. it may make sense to choose an unusual orientation for the axes if it makes sense for the problem. e. a football has rotational symmetry about a central axis, so cylindrical coordinates would work best. the z - axis should align with the axis of the ball. the origin could be the center of the ball or perhaps one of the ends. the position of the x - axis is arbitrary. 2. 61 which coordinate system is most appropriate for creating a star map, as viewed from earth ( see the following figure )? how should we orient the coordinate axes? | openstax_calculus_volume_3_-_web | [
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the center of the ball or perhaps one of the ends. the position of the x - axis is arbitrary. 2. 61 which coordinate system is most appropriate for creating a star map, as viewed from earth ( see the following figure )? how should we orient the coordinate axes? | openstax_calculus_volume_3_-_web | [
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2. 7 • cylindrical and spherical coordinates 223 section 2. 7 exercises use the following figure as an aid in identifying the relationship between the rectangular, cylindrical, and spherical coordinate systems. for the following exercises, the cylindrical coordinates of a point are given. find the rectangular coordinates of the point. 363. 364. 365. 366. for the following exercises, the rectangular coordinates of a point are given. find the cylindrical coordinates of the point. 367. 368. 369. 370. for the following exercises, the equation of a surface in cylindrical coordinates is given. find the equation of the surface in rectangular coordinates. identify and graph the surface. 371. [ t ] 372. [ t ] 373. [ t ] 374. [ t ] 375. [ t ] 376. [ t ] 377. [ t ] 378. [ t ] for the following exercises, the equation of a surface in rectangular coordinates is given. find the equation of the surface in cylindrical coordinates. 379. 380. 381. 382. 383. 384. for the following exercises, the spherical coordinates of a point are given. find the rectangular coordinates of the point. 385. 386. 387. 388. 224 2 • vectors in space access for free at openstax. org for the following exercises, the rectangular coordinates of a point are given. find the spherical coordinates of the point. express the measure of the angles in degrees rounded to the nearest integer. 389. 390. 391. 392. for the following exercises, the equation of a surface in spherical coordinates is given. find the equation of the surface in rectangular coordinates. identify and graph the surface. 393. [ t ] 394. [ t ] 395. [ t ] 396. [ t ] 397. [ t ] 398. [ t ] for the following exercises, the equation of a surface in rectangular coordinates is given. find the equation of the surface in spherical coordinates. identify the surface. 399. 400. 401. 402. for the following exercises, the cylindrical coordinates of a point are given. find its associated spherical coordinates, with the measure of the angle | openstax_calculus_volume_3_-_web | [
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the equation of a surface in rectangular coordinates is given. find the equation of the surface in spherical coordinates. identify the surface. 399. 400. 401. 402. for the following exercises, the cylindrical coordinates of a point are given. find its associated spherical coordinates, with the measure of the angle in radians rounded to four decimal places. 403. [ t ] 404. [ t ] 405. 406. for the following exercises, the spherical coordinates of a point are given. find its associated cylindrical coordinates. 407. 408. 409. 410. for the following exercises, find the most suitable system of coordinates to describe the solids. 411. the solid situated in the first octant with a vertex at the origin and enclosed by a cube of edge length where 412. a spherical shell determined by the region between two concentric spheres centered at the origin, of radii of and respectively, where 413. a solid inside sphere and outside cylinder 414. a cylindrical shell of height determined by the region between two cylinders with the same center, parallel rulings, and radii of and respectively 415. [ t ] use a cas to graph the region between elliptic paraboloid and cone then describe the region in cylindrical coordinates. 416. [ t ] use a cas to graph in spherical coordinates the “ ice cream - cone region ” situated above the xy - plane between sphere and elliptical cone | openstax_calculus_volume_3_-_web | [
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2. 7 • cylindrical and spherical coordinates 225 417. washington, dc, is located at n and w ( see the following figure ). assume the radius of earth is mi. express the location of washington, dc, in spherical coordinates. 418. san francisco is located at and assume the radius of earth is mi. express the location of san francisco in spherical coordinates. 419. find the latitude and longitude of rio de janeiro if its spherical coordinates are 420. find the latitude and longitude of berlin if its spherical coordinates are 421. [ t ] consider the torus of equation where a. write the equation of the torus in spherical coordinates. b. if the surface is called a horn torus. show that the equation of a horn torus in spherical coordinates is c. use a cas to graph the horn torus with in spherical coordinates. 226 2 • vectors in space access for free at openstax. org 422. [ t ] the “ bumpy sphere ” with an equation in spherical coordinates is with and where and are positive numbers and and are positive integers, may be used in applied mathematics to model tumor growth. a. show that the “ bumpy sphere ” is contained inside a sphere of equation find the values of and at which the two surfaces intersect. b. use a cas to graph the surface for and along with sphere c. find the equation of the intersection curve of the surface at b. with the cone graph the intersection curve in the plane of intersection. | openstax_calculus_volume_3_-_web | [
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2. 7 • cylindrical and spherical coordinates 227 chapter review key terms component a scalar that describes either the vertical or horizontal direction of a vector coordinate plane a plane containing two of the three coordinate axes in the three - dimensional coordinate system, named by the axes it contains : the xy - plane, xz - plane, or the yz - plane cross product where and cylinder a set of lines parallel to a given line passing through a given curve cylindrical coordinate system a way to describe a location in space with an ordered triple where represents the polar coordinates of the point ’ s projection in the xy - plane, and represents the point ’ s projection onto the z - axis determinant a real number associated with a square matrix direction angles the angles formed by a nonzero vector and the coordinate axes direction cosines the cosines of the angles formed by a nonzero vector and the coordinate axes direction vector a vector parallel to a line that is used to describe the direction, or orientation, of the line in space dot product or scalar product where and ellipsoid a three - dimensional surface described by an equation of the form all traces of this surface are ellipses elliptic cone a three - dimensional surface described by an equation of the form traces of this surface include ellipses and intersecting lines elliptic paraboloid a three - dimensional surface described by an equation of the form traces of this surface include ellipses and parabolas equivalent vectors vectors that have the same magnitude and the same direction general form of the equation of a plane an equation in the form where is a normal vector of the plane, is a point on the plane, and hyperboloid of one sheet a three - dimensional surface described by an equation of the form traces of this surface include ellipses and hyperbolas hyperboloid of two sheets a three - dimensional surface described by an equation of the form traces of this surface include ellipses and hyperbolas initial point the starting point of a vector magnitude the length of a vector normal vector a vector perpendicular to a plane normalization using scalar multiplication to find a unit vector with a given direction octants the eight regions of space created by the coordinate planes orthogonal vectors vectors that form a | openstax_calculus_volume_3_-_web | [
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##ses and hyperbolas initial point the starting point of a vector magnitude the length of a vector normal vector a vector perpendicular to a plane normalization using scalar multiplication to find a unit vector with a given direction octants the eight regions of space created by the coordinate planes orthogonal vectors vectors that form a right angle when placed in standard position parallelepiped a three - dimensional prism with six faces that are parallelograms parallelogram method a method for finding the sum of two vectors ; position the vectors so they share the same initial point ; the vectors then form two adjacent sides of a parallelogram ; the sum of the vectors is the diagonal of that parallelogram parametric equations of a line the set of equations and describing the line with direction vector passing through point quadric surfaces surfaces in three dimensions having the property that the traces of the surface are conic sections ( ellipses, hyperbolas, and parabolas ) right - hand rule a common way to define the orientation of the three - dimensional coordinate system ; when the right hand is curved around the z - axis in such a way that the fingers curl from the positive x - axis to the positive y - axis, the thumb points in the direction of the positive z - axis rulings parallel lines that make up a cylindrical surface scalar a real number scalar equation of a plane the equation used to describe a plane containing point with normal vector or its alternate form where scalar multiplication a vector operation that defines the product of a scalar and a vector scalar projection the magnitude of the vector projection of a vector 228 2 • chapter review access for free at openstax. org skew lines two lines that are not parallel but do not intersect sphere the set of all points equidistant from a given point known as the center spherical coordinate system a way to describe a location in space with an ordered triple where is the distance between and the origin is the same angle used to describe the location in cylindrical coordinates, and is the angle formed by the positive z - axis and line segment where is the origin and standard equation of a sphere describes a sphere with center and radius standard unit vectors unit vectors along the coordinate axes : standard - position vector a vector with | openstax_calculus_volume_3_-_web | [
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the same angle used to describe the location in cylindrical coordinates, and is the angle formed by the positive z - axis and line segment where is the origin and standard equation of a sphere describes a sphere with center and radius standard unit vectors unit vectors along the coordinate axes : standard - position vector a vector with initial point symmetric equations of line the equations describing the line with direction vector passing through point terminal point the endpoint of a vector three - dimensional rectangular coordinate system a coordinate system defined by three lines that intersect at right angles ; every point in space is described by an ordered triple that plots its location relative to the defining axes torque the effect of a force that causes an object to rotate trace the intersection of a three - dimensional surface with a coordinate plane triangle inequality the length of any side of a triangle is less than the sum of the lengths of the other two sides triangle method a method for finding the sum of two vectors ; position the vectors so the terminal point of one vector is the initial point of the other ; these vectors then form two sides of a triangle ; the sum of the vectors is the vector that forms the third side ; the initial point of the sum is the initial point of the first vector ; the terminal point of the sum is the terminal point of the second vector triple scalar product the dot product of a vector with the cross product of two other vectors : unit vector a vector with margnitude vector a mathematical object that has both magnitude and direction vector addition a vector operation that defines the sum of two vectors vector difference the vector difference is defined as vector equation of a line the equation used to describe a line with direction vector passing through point where is the position vector of point vector equation of a plane the equation where is a given point in the plane, is any point in the plane, and is a normal vector of the plane vector product the cross product of two vectors vector projection the component of a vector that follows a given direction vector sum the sum of two vectors, and can be constructed graphically by placing the initial point of at the terminal point of then the vector sum is the vector with an initial point that coincides with the initial point of and with a terminal point that coincides with the terminal point of work done by a force | openstax_calculus_volume_3_-_web | [
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the sum of two vectors, and can be constructed graphically by placing the initial point of at the terminal point of then the vector sum is the vector with an initial point that coincides with the initial point of and with a terminal point that coincides with the terminal point of work done by a force work is generally thought of as the amount of energy it takes to move an object ; if we represent an applied force by a vector f and the displacement of an object by a vector s, then the work done by the force is the dot product of f and s. zero vector the vector with both initial point and terminal point key equations distance between two points in space : sphere with center and radius r : dot product of u and v cosine of the angle formed by and vector projection of onto 2 • chapter review 229 scalar projection of onto work done by a force f to move an object through displacement vector the cross product of two vectors in terms of the unit vectors vector equation of a line parametric equations of a line vector equation of a plane scalar equation of a plane distance between a plane and a point key concepts | openstax_calculus_volume_3_-_web | [
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2. 1 vectors in the plane • vectors are used to represent quantities that have both magnitude and direction. • we can add vectors by using the parallelogram method or the triangle method to find the sum. we can multiply a vector by a scalar to change its length or give it the opposite direction. • subtraction of vectors is defined in terms of adding the negative of the vector. • a vector is written in component form as • the magnitude of a vector is a scalar : • a unit vector has magnitude and can be found by dividing a vector by its magnitude : the standard unit vectors are a vector can be expressed in terms of the standard unit vectors as • vectors are often used in physics and engineering to represent forces and velocities, among other quantities. | openstax_calculus_volume_3_-_web | [
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2. 2 vectors in three dimensions • the three - dimensional coordinate system is built around a set of three axes that intersect at right angles at a single point, the origin. ordered triples are used to describe the location of a point in space. • the distance between points and is given by the formula • in three dimensions, the equations describe planes that are parallel to the coordinate planes. • the standard equation of a sphere with center and radius is • in three dimensions, as in two, vectors are commonly expressed in component form, or in terms of the standard unit vectors, • properties of vectors in space are a natural extension of the properties for vectors in a plane. let and be vectors, and let be a scalar. scalar multiplication : vector addition : vector subtraction : vector magnitude : unit vector in the direction of v : 230 2 • chapter review access for free at openstax. org | openstax_calculus_volume_3_-_web | [
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2. 3 the dot product • the dot product, or scalar product, of two vectors and is • the dot product satisfies the following properties : • the dot product of two vectors can be expressed, alternatively, as this form of the dot product is useful for finding the measure of the angle formed by two vectors. • vectors and are orthogonal if • the angles formed by a nonzero vector and the coordinate axes are called the direction angles for the vector. the cosines of these angles are known as the direction cosines. • the vector projection of onto is the vector the magnitude of this vector is known as the scalar projection of onto, given by • work is done when a force is applied to an object, causing displacement. when the force is represented by the vector f and the displacement is represented by the vector s, then the work done w is given by the formula | openstax_calculus_volume_3_-_web | [
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2. 4 the cross product • the cross product of two vectors and is a vector orthogonal to both and its length is given by where is the angle between and its direction is given by the right - hand rule. • the algebraic formula for calculating the cross product of two vectors, is • the cross product satisfies the following properties for vectors and scalar • the cross product of vectors and is the determinant • if vectors and form adjacent sides of a parallelogram, then the area of the parallelogram is given by • the triple scalar product of vectors and is • the volume of a parallelepiped with adjacent edges given by vectors is • if the triple scalar product of vectors is zero, then the vectors are coplanar. the converse is also true : if the vectors are coplanar, then their triple scalar product is zero. • the cross product can be used to identify a vector orthogonal to two given vectors or to a plane. • torque measures the tendency of a force to produce rotation about an axis of rotation. if force is acting at a distance from the axis, then torque is equal to the cross product of and | openstax_calculus_volume_3_-_web | [
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2. 5 equations of lines and planes in space • in three dimensions, the direction of a line is described by a direction vector. the vector equation of a line with direction vector passing through point is where is the position vector of point this equation can be rewritten to form the parametric equations of the line : and the line can also be described with the symmetric equations • let be a line in space passing through point with direction vector if is any point not on then the distance from to is • in three dimensions, two lines may be parallel but not equal, equal, intersecting, or skew. • given a point and vector the set of all points satisfying equation forms a plane. equation 2 • chapter review 231 is known as the vector equation of a plane. • the scalar equation of a plane containing point with normal vector is this equation can be expressed as where this form of the equation is sometimes called the general form of the equation of a plane. • suppose a plane with normal vector n passes through point the distance from the plane to point not in the plane is given by • the normal vectors of parallel planes are parallel. when two planes intersect, they form a line. • the measure of the angle between two intersecting planes can be found using the equation : where and are normal vectors to the planes. • the distance from point to plane is given by | openstax_calculus_volume_3_-_web | [
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2. 6 quadric surfaces • a set of lines parallel to a given line passing through a given curve is called a cylinder, or a cylindrical surface. the parallel lines are called rulings. • the intersection of a three - dimensional surface and a plane is called a trace. to find the trace in the xy -, yz -, or xz - planes, set respectively. • quadric surfaces are three - dimensional surfaces with traces composed of conic sections. every quadric surface can be expressed with an equation of the form • to sketch the graph of a quadric surface, start by sketching the traces to understand the framework of the surface. • important quadric surfaces are summarized in figure 2. 87 and figure 2. 88. | openstax_calculus_volume_3_-_web | [
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2. 7 cylindrical and spherical coordinates • in the cylindrical coordinate system, a point in space is represented by the ordered triple where represents the polar coordinates of the point ’ s projection in the xy - plane and represents the point ’ s projection onto the z - axis. • to convert a point from cylindrical coordinates to cartesian coordinates, use equations and • to convert a point from cartesian coordinates to cylindrical coordinates, use equations and • in the spherical coordinate system, a point in space is represented by the ordered triple where is the distance between and the origin is the same angle used to describe the location in cylindrical coordinates, and is the angle formed by the positive z - axis and line segment where is the origin and • to convert a point from spherical coordinates to cartesian coordinates, use equations and • to convert a point from cartesian coordinates to spherical coordinates, use equations and • to convert a point from spherical coordinates to cylindrical coordinates, use equations and • to convert a point from cylindrical coordinates to spherical coordinates, use equations and 232 2 • chapter review access for free at openstax. org review exercises for the following exercises, determine whether the statement is true or false. justify the answer with a proof or a counterexample. 423. for vectors and and any given scalar 424. for vectors and and any given scalar 425. the symmetric equation for the line of intersection between two planes and is given by 426. if then is perpendicular to for the following exercises, use the given vectors to find the quantities. 427. a. b. c. d. 428. a. b. c. d. e. 429. find the values of such that vectors and are orthogonal. for the following exercises, find the unit vectors. 430. find the unit vector that has the same direction as vector that begins at and ends at 431. find the unit vector that has the same direction as vector that begins at and ends at for the following exercises, find the area or volume of the given shapes. 432. the parallelogram spanned by vectors 433. the parallelepiped formed by and for the following exercises, find the vector and parametric equations of the line with the given | openstax_calculus_volume_3_-_web | [
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that begins at and ends at for the following exercises, find the area or volume of the given shapes. 432. the parallelogram spanned by vectors 433. the parallelepiped formed by and for the following exercises, find the vector and parametric equations of the line with the given properties. 434. the line that passes through point that is parallel to vector 435. the line that passes through points and for the following exercises, find the equation of the plane with the given properties. 436. the plane that passes through point and has normal vector 437. the plane that passes through points for the following exercises, find the traces for the surfaces in planes then, describe and draw the surfaces. 438. 439. 2 • chapter review 233 for the following exercises, write the given equation in cylindrical coordinates and spherical coordinates. 440. 441. for the following exercises, convert the given equations from cylindrical or spherical coordinates to rectangular coordinates. identify the given surface. 442. 443. for the following exercises, consider a small boat crossing a river. 444. if the boat velocity is km / h due north in still water and the water has a current of km / h due west ( see the following figure ), what is the velocity of the boat relative to shore? what is the angle that the boat is actually traveling? 445. when the boat reaches the shore, two ropes are thrown to people to help pull the boat ashore. one rope is at an angle of and the other is at if the boat must be pulled straight and at a force of find the magnitude of force for each rope ( see the following figure ). 446. an airplane is flying in the direction of 52° east of north with a speed of 450 mph. a strong wind has a bearing 33° east of north with a speed of 50 mph. what is the resultant ground speed and bearing of the airplane? 447. calculate the work done by moving a particle from position to along a straight line with a force the following problems consider your unsuccessful attempt to take the tire off your car using a wrench to loosen the bolts. assume the wrench is m long and you are able to apply a 200 - n | openstax_calculus_volume_3_-_web | [
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##7. calculate the work done by moving a particle from position to along a straight line with a force the following problems consider your unsuccessful attempt to take the tire off your car using a wrench to loosen the bolts. assume the wrench is m long and you are able to apply a 200 - n force. 448. because your tire is flat, you are only able to apply your force at a angle. what is the torque at the center of the bolt? assume this force is not enough to loosen the bolt. 449. someone lends you a tire jack and you are now able to apply a 200 - n force at an angle. is your resulting torque going to be more or less? what is the new resulting torque at the center of the bolt? assume this force is not enough to loosen the bolt. 234 2 • chapter review access for free at openstax. org figure 3. 1 halley ’ s comet appeared in view of earth in 1986 and will appear again in 2061. chapter outline 3. 1 vector - valued functions and space curves 3. 2 calculus of vector - valued functions 3. 3 arc length and curvature | openstax_calculus_volume_3_-_web | [
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3. 4 motion in space introduction in 1705, using sir isaac newton ’ s new laws of motion, the astronomer edmond halley made a prediction. he stated that comets that had appeared in 1531, 1607, and 1682 were actually the same comet and that it would reappear in 1758. halley was proved to be correct, although he did not live to see it. however, the comet was later named in his honor. halley ’ s comet follows an elliptical path through the solar system, with the sun appearing at one focus of the ellipse. this motion is predicted by johannes kepler ’ s first law of planetary motion, which we mentioned briefly in the introduction to parametric equations and polar coordinates. in example 3. 15, we show how to use kepler ’ s third law of planetary motion along with the calculus of vector - valued functions to find the average distance of halley ’ s comet from the sun. vector - valued functions provide a useful method for studying various curves both in the plane and in three - dimensional space. we can apply this concept to calculate the velocity, acceleration, arc length, and curvature of an object ’ s trajectory. in this chapter, we examine these methods and show how they are used. | openstax_calculus_volume_3_-_web | [
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3. 1 vector - valued functions and space curves learning objectives 3. 1. 1 write the general equation of a vector - valued function in component form and unit - vector form. 3. 1. 2 recognize parametric equations for a space curve. 3. 1. 3 describe the shape of a helix and write its equation. 3. 1. 4 define the limit of a vector - valued function. our study of vector - valued functions combines ideas from our earlier examination of single - variable calculus with our description of vectors in three dimensions from the preceding chapter. in this section we extend concepts from earlier chapters and also examine new ideas concerning curves in three - dimensional space. these definitions and theorems support the presentation of material in the rest of this chapter and also in the remaining chapters of the text. definition of a vector - valued function our first step in studying the calculus of vector - valued functions is to define what exactly a vector - valued function is. we 3 vector - valued functions 3 • introduction 235 can then look at graphs of vector - valued functions and see how they define curves in both two and three dimensions. definition a vector - valued function is a function of the form where the component functions f, g, and h, are real - valued functions of the parameter t. vector - valued functions are also written in the form in both cases, the first form of the function defines a two - dimensional vector - valued function ; the second form describes a three - dimensional vector - valued function. the parameter t can lie between two real numbers : another possibility is that the value of t might take on all real numbers. last, the component functions themselves may have domain restrictions that enforce restrictions on the value of t. we often use t as a parameter because t can represent time. example 3. 1 evaluating vector - valued functions and determining domains for each of the following vector - valued functions, evaluate do any of these functions have domain restrictions? a. b. solution a. to calculate each of the function values, substitute the appropriate value of t into the function : to determine whether this function has any domain restrictions, consider the component functions separately. the first component function is and the second component function is neither of these functions has a domain restriction, | openstax_calculus_volume_3_-_web | [
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. b. solution a. to calculate each of the function values, substitute the appropriate value of t into the function : to determine whether this function has any domain restrictions, consider the component functions separately. the first component function is and the second component function is neither of these functions has a domain restriction, so the domain of is all real numbers. b. to calculate each of the function values, substitute the appropriate value of t into the function : to determine whether this function has any domain restrictions, consider the component functions separately. the first component function is the second component function is and the third component function is the first two functions are not defined for odd multiples of so the function is not defined ( 3. 1 ) ( 3. 2 ) 236 3 • vector - valued functions access for free at openstax. org for odd multiples of therefore, where n is any integer. | openstax_calculus_volume_3_-_web | [
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3. 1 for the vector - valued function evaluate does this function have any domain restrictions? example 3. 1 illustrates an important concept. the domain of a vector - valued function consists of real numbers. the domain can be all real numbers or a subset of the real numbers. the range of a vector - valued function consists of vectors. each real number in the domain of a vector - valued function is mapped to either a two - or a three - dimensional vector. graphing vector - valued functions recall that a plane vector consists of two quantities : direction and magnitude. given any point in the plane ( the initial point ), if we move in a specific direction for a specific distance, we arrive at a second point. this represents the terminal point of the vector. we calculate the components of the vector by subtracting the coordinates of the initial point from the coordinates of the terminal point. a vector is considered to be in standard position if the initial point is located at the origin. when graphing a vector - valued function, we typically graph the vectors in the domain of the function in standard position, because doing so guarantees the uniqueness of the graph. this convention applies to the graphs of three - dimensional vector - valued functions as well. the graph of a vector - valued function of the form consists of the set of all and the path it traces is called a plane curve. the graph of a vector - valued function of the form consists of the set of all and the path it traces is called a space curve. any representation of a plane curve or space curve using a vector - valued function is called a vector parameterization of the curve. example 3. 2 graphing a vector - valued function create a graph of each of the following vector - valued functions : a. the plane curve represented by b. the plane curve represented by c. the space curve represented by solution a. as with any graph, we start with a table of values. we then graph each of the vectors in the second column of the table in standard position and connect the terminal points of each vector to form a curve ( figure 3. 2 ). this curve turns out to be an ellipse centered at the origin. t t 0 table 3. 1 table | openstax_calculus_volume_3_-_web | [
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. we then graph each of the vectors in the second column of the table in standard position and connect the terminal points of each vector to form a curve ( figure 3. 2 ). this curve turns out to be an ellipse centered at the origin. t t 0 table 3. 1 table of values for | openstax_calculus_volume_3_-_web | [
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3. 1 • vector - valued functions and space curves 237 figure 3. 2 the graph of the first vector - valued function is an ellipse. b. the table of values for is as follows : t t 0 table 3. 2 table of values for the graph of this curve is also an ellipse centered at the origin. 238 3 • vector - valued functions access for free at openstax. org figure 3. 3 the graph of the second vector - valued function is also an ellipse. c. we go through the same procedure for a three - dimensional vector function. t t 0 table 3. 3 table of values for the values then repeat themselves, except for the fact that the coefficient of k is always increasing ( figure 3. 4 ). this curve is called a helix. notice that if the k component is eliminated, then the function becomes which is a unit circle centered at the origin. figure 3. 4 the graph of the third vector - valued function is a helix. | openstax_calculus_volume_3_-_web | [
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3. 1 • vector - valued functions and space curves 239 you may notice that the graphs in parts a. and b. are identical. this happens because the function describing curve b is a so - called reparameterization of the function describing curve a. in fact, any curve has an infinite number of reparameterizations ; for example, we can replace t with in any of the three previous curves without changing the shape of the curve. the interval over which t is defined may change, but that is all. we return to this idea later in this chapter when we study arc - length parameterization. as mentioned, the name of the shape of the curve of the graph in example 3. 2c. is a helix ( figure 3. 4 ). the curve resembles a spring, with a circular cross - section looking down along the z - axis. it is possible for a helix to be elliptical in cross - section as well. for example, the vector - valued function describes an elliptical helix. the projection of this helix into the is an ellipse. last, the arrows in the graph of this helix indicate the orientation of the curve as t progresses from 0 to | openstax_calculus_volume_3_-_web | [
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3. 2 create a graph of the vector - valued function at this point, you may notice a similarity between vector - valued functions and parameterized curves. indeed, given a vector - valued function we can define and if a restriction exists on the values of t ( for example, t is restricted to the interval for some constants then this restriction is enforced on the parameter. the graph of the parameterized function would then agree with the graph of the vector - valued function, except that the vector - valued graph would represent vectors rather than points. since we can parameterize a curve defined by a function it is also possible to represent an arbitrary plane curve by a vector - valued function. limits and continuity of a vector - valued function we now take a look at the limit of a vector - valued function. this is important to understand to study the calculus of vector - valued functions. definition a vector - valued function r approaches the limit l as t approaches a, written provided this is a rigorous definition of the limit of a vector - valued function. in practice, we use the following theorem : theorem 3. 1 limit of a vector - valued function let f, g, and h be functions of t. then the limit of the vector - valued function as t approaches a is given by provided the limits exist. similarly, the limit of the vector - valued function as t approaches a is given by provided the limits exist. in the following example, we show how to calculate the limit of a vector - valued function. ( 3. 3 ) ( 3. 4 ) 240 3 • vector - valued functions access for free at openstax. org example 3. 3 evaluating the limit of a vector - valued function for each of the following vector - valued functions, calculate for a. b. solution a. use equation 3. 3 and substitute the value into the two component expressions : b. use equation 3. 4 and substitute the value into the three component expressions : 3. 3 calculate for the function now that we know how to calculate the limit of a vector - valued function, we can define continuity at a point for such a function. definition let f, g, and h be functions of t. then, the vector - valued function is continuous | openstax_calculus_volume_3_-_web | [
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component expressions : 3. 3 calculate for the function now that we know how to calculate the limit of a vector - valued function, we can define continuity at a point for such a function. definition let f, g, and h be functions of t. then, the vector - valued function is continuous at point if the following three conditions hold : 1. exists 2. exists 3. similarly, the vector - valued function is continuous at point if the following three conditions hold : 1. exists 2. exists 3. | openstax_calculus_volume_3_-_web | [
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3. 1 • vector - valued functions and space curves 241 section 3. 1 exercises 1. give the component functions and for the vector - valued function 2. given find the following values ( if possible ). a. b. c. 3. sketch the curve of the vector - valued function and give the orientation of the curve. sketch asymptotes as a guide to the graph. 4. evaluate 5. given the vector - valued function find the following values : a. b. c. is continuous at d. graph 6. given the vector - valued function find the following values : a. b. c. is continuous at d. 7. let find the following values : a. b. c. is continuous at find the limit of the following vector - valued functions at the indicated value of t. 8. 9. for 10. ∞ 11. 12. 13. ∞ for 14. describe the curve defined by the vector - valued function find the domain of the vector - valued functions. 15. domain : 16. domain : 17. domain : let and use it to answer the following questions. 18. for what values of t is continuous? 19. sketch the graph of 20. find the domain of 21. for what values of t is continuous? 242 3 • vector - valued functions access for free at openstax. org eliminate the parameter t, write the equation in cartesian coordinates, then sketch the graphs of the vector - valued functions. 22. ( hint : let and solve the first equation for x in terms of t and substitute this result into the second equation. ) 23. 24. 25. 26. use a graphing utility to sketch each of the following vector - valued functions : 27. [ t ] 28. [ t ] 29. [ t ] 30. clockwise and counterclockwise 31. from left to right 32. the line through p and q where p is and q is consider the curve described by the vector - valued function 33. what is the initial point of the path corresponding to 34. what is ∞ 35. [ t ] use technology to sketch the curve. 36. eliminate the parameter t to show that where 37. [ t ] let use technology to | openstax_calculus_volume_3_-_web | [
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q is consider the curve described by the vector - valued function 33. what is the initial point of the path corresponding to 34. what is ∞ 35. [ t ] use technology to sketch the curve. 36. eliminate the parameter t to show that where 37. [ t ] let use technology to graph the curve ( called the roller - coaster curve ) over the interval choose at least two views to determine the peaks and valleys. 38. [ t ] use the result of the preceding problem to construct an equation of a roller coaster with a steep drop from the peak and steep incline from the “ valley. ” then, use technology to graph the equation. 39. use the results of the preceding two problems to construct an equation of a path of a roller coaster with more than two turning points ( peaks and valleys ). 40. a. graph the curve using two viewing angles of your choice to see the overall shape of the curve. b. does the curve resemble a “ slinky ”? c. what changes to the equation should be made to increase the number of coils of the slinky? | openstax_calculus_volume_3_-_web | [
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3. 2 calculus of vector - valued functions learning objectives 3. 2. 1 write an expression for the derivative of a vector - valued function. 3. 2. 2 find the tangent vector at a point for a given position vector. 3. 2. 3 find the unit tangent vector at a point for a given position vector and explain its significance. 3. 2. 4 calculate the definite integral of a vector - valued function. to study the calculus of vector - valued functions, we follow a similar path to the one we took in studying real - valued functions. first, we define the derivative, then we examine applications of the derivative, then we move on to defining integrals. however, we will find some interesting new ideas along the way as a result of the vector nature of these functions and the properties of space curves. derivatives of vector - valued functions now that we have seen what a vector - valued function is and how to take its limit, the next step is to learn how to differentiate a vector - valued function. the definition of the derivative of a vector - valued function is nearly identical to the definition of a real - valued function of one variable. however, because the range of a vector - valued function consists of | openstax_calculus_volume_3_-_web | [
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3. 2 • calculus of vector - valued functions 243 vectors, the same is true for the range of the derivative of a vector - valued function. definition the derivative of a vector - valued function is provided the limit exists. if exists, then r is differentiable at t. if exists for all t in an open interval then r is differentiable over the interval for the function to be differentiable over the closed interval the following two limits must exist as well : many of the rules for calculating derivatives of real - valued functions can be applied to calculating the derivatives of vector - valued functions as well. recall that the derivative of a real - valued function can be interpreted as the slope of a tangent line or the instantaneous rate of change of the function. the derivative of a vector - valued function can be understood to be an instantaneous rate of change as well ; for example, when the function represents the position of an object at a given point in time, the derivative represents its velocity at that same point in time. we now demonstrate taking the derivative of a vector - valued function. example 3. 4 finding the derivative of a vector - valued function use the definition to calculate the derivative of the function solution let ’ s use equation 3. 5 : | openstax_calculus_volume_3_-_web | [
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3. 4 use the definition to calculate the derivative of the function notice that in the calculations in example 3. 4, we could also obtain the answer by first calculating the derivative of each component function, then putting these derivatives back into the vector - valued function. this is always true for calculating the derivative of a vector - valued function, whether it is in two or three dimensions. we state this in the following theorem. the proof of this theorem follows directly from the definitions of the limit of a vector - valued function and the derivative of a vector - valued function. ( 3. 5 ) 244 3 • vector - valued functions access for free at openstax. org theorem 3. 2 differentiation of vector - valued functions let f, g, and h be differentiable functions of t. i. if then ii. if then example 3. 5 calculating the derivative of vector - valued functions use differentiation of vector - valued functions to calculate the derivative of each of the following functions. a. b. c. solution we use differentiation of vector - valued functions and what we know about differentiating functions of one variable. a. the first component of is the second component is we have and so the theorem gives b. the first component is and the second component is we have and so we obtain c. the first component of is the second component is and the third component is we have and so the theorem gives | openstax_calculus_volume_3_-_web | [
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3. 5 calculate the derivative of the function we can extend to vector - valued functions the properties of the derivative that we presented in the introduction to derivatives ( http : / / openstax. org / books / calculus - volume - 1 / pages / 3 - introduction ). in particular, the constant multiple rule, the sum and difference rules, the product rule, and the chain rule all extend to vector - valued functions. however, in the case of the product rule, there are actually three extensions : ( 1 ) for a real - valued function multiplied by a vector - valued function, ( 2 ) for the dot product of two vector - valued functions, and ( 3 ) for the cross product of two vector - valued functions. theorem 3. 3 properties of the derivative of vector - valued functions let r and u be differentiable vector - valued functions of t, let f be a differentiable real - valued function of t, and let c be a scalar. | openstax_calculus_volume_3_-_web | [
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3. 2 • calculus of vector - valued functions 245 proof the proofs of the first two properties follow directly from the definition of the derivative of a vector - valued function. the third property can be derived from the first two properties, along with the product rule from the introduction to derivatives ( http : / / openstax. org / books / calculus - volume - 1 / pages / 3 - introduction ). let then to prove property iv. let and then the proof of property v. is similar to that of property iv. property vi. can be proved using the chain rule. last, property vii. follows from property iv : now for some examples using these properties. example 3. 6 using the properties of derivatives of vector - valued functions given the vector - valued functions and calculate each of the following derivatives using the properties of the derivative of vector - valued functions. a. b. solution a. we have and therefore, according to property iv. : b. first, we need to adapt property v. for this problem : 246 3 • vector - valued functions access for free at openstax. org recall that the cross product of any vector with itself is zero. furthermore, represents the second derivative of therefore, | openstax_calculus_volume_3_-_web | [
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3. 6 given the vector - valued functions and calculate and tangent vectors and unit tangent vectors recall from the introduction to derivatives ( http : / / openstax. org / books / calculus - volume - 1 / pages / 3 - introduction ) that the derivative at a point can be interpreted as the slope of the tangent line to the graph at that point. in the case of a vector - valued function, the derivative provides a tangent vector to the curve represented by the function. consider the vector - valued function the derivative of this function is if we substitute the value into both functions we get the graph of this function appears in figure 3. 5, along with the vectors and figure 3. 5 the tangent line at a point is calculated from the derivative of the vector - valued function notice that the vector is tangent to the circle at the point corresponding to this is an example of a tangent vector to the plane curve defined by | openstax_calculus_volume_3_-_web | [
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3. 2 • calculus of vector - valued functions 247 definition let c be a curve defined by a vector - valued function r, and assume that exists when a tangent vector v at is any vector such that, when the tail of the vector is placed at point on the graph, vector v is tangent to curve c. vector is an example of a tangent vector at point furthermore, assume that the principal unit tangent vector at t is defined to be provided the unit tangent vector is exactly what it sounds like : a unit vector that is tangent to the curve. to calculate a unit tangent vector, first find the derivative second, calculate the magnitude of the derivative. the third step is to divide the derivative by its magnitude. example 3. 7 finding a unit tangent vector find the unit tangent vector for each of the following vector - valued functions : a. b. solution a. b. | openstax_calculus_volume_3_-_web | [
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3. 7 find the unit tangent vector for the vector - valued function integrals of vector - valued functions we introduced antiderivatives of real - valued functions in antiderivatives ( http : / / openstax. org / books / calculus - volume - 1 / pages / 4 - 10 - antiderivatives ) and definite integrals of real - valued functions in the definite integral ( http : / / openstax. org / books / calculus - volume - 2 / pages / 1 - 2 - the - definite - integral ). each of these concepts can be extended to vector - valued functions. also, just as we can calculate the derivative of a vector - valued function by differentiating the component functions separately, we can calculate the antiderivative in the same manner. furthermore, the fundamental theorem of calculus applies to vector - valued functions as well. the antiderivative of a vector - valued function appears in applications. for example, if a vector - valued function represents the velocity of an object at time t, then its antiderivative represents position. or, if the function represents the ( 3. 6 ) 248 3 • vector - valued functions access for free at openstax. org acceleration of the object at a given time, then the antiderivative represents its velocity. definition let f, g, and h be integrable real - valued functions over the closed interval 1. the indefinite integral of a vector - valued function is the definite integral of a vector - valued function is 2. the indefinite integral of a vector - valued function is the definite integral of the vector - valued function is since the indefinite integral of a vector - valued function involves indefinite integrals of the component functions, each of these component integrals contains an integration constant. they can all be different. for example, in the two - dimensional case, we can have where f and g are antiderivatives of f and g, respectively. then where therefore, the integration constant becomes a constant vector. example 3. 8 integrating vector - valued functions calculate each of the following integrals : a. b. c. solution a. we use the first part of the definition of the integral of | openstax_calculus_volume_3_-_web | [
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##ivatives of f and g, respectively. then where therefore, the integration constant becomes a constant vector. example 3. 8 integrating vector - valued functions calculate each of the following integrals : a. b. c. solution a. we use the first part of the definition of the integral of a space curve : ( 3. 7 ) ( 3. 8 ) ( 3. 9 ) ( 3. 10 ) 3. 2 • calculus of vector - valued functions 249 b. first calculate next, substitute this back into the integral and integrate : c. use the second part of the definition of the integral of a space curve : 3. 8 calculate the following integral : 250 3 • vector - valued functions access for free at openstax. org section 3. 2 exercises compute the derivatives of the vector - valued functions. 41. 42. 43. a sketch of the graph is shown here. notice the varying periodic nature of the graph. 44. 45. 46. 47. 48. 49. 50. for the following problems, find a tangent vector at the indicated value of t. 51. 52. 53. 54. | openstax_calculus_volume_3_-_web | [
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3. 2 • calculus of vector - valued functions 251 find the unit tangent vector for the following parameterized curves. 55.. two views of this curve are presented here : 56. 57. 58. let and here is the graph of the function : find the following. 59. 60. 61. 62. compute the first, second, and third derivatives of 63. find 252 3 • vector - valued functions access for free at openstax. org 64. the acceleration function, initial velocity, and initial position of a particle are find 65. the position vector of a particle is a. graph the position function and display a view of the graph that illustrates the asymptotic behavior of the function. b. find the velocity as t approaches but is not equal to ( if it exists ). 66. find the velocity and the speed of a particle with the position function the speed of a particle is the magnitude of the velocity and is represented by a particle moves on a circular path of radius b according to the function where is the angular velocity, 67. find the velocity function and show that is always orthogonal to 68. show that the speed of the particle is proportional to the angular velocity. 69. evaluate given 70. find the antiderivative of that satisfies the initial condition 71. evaluate 72. an object starts from rest at point and moves with an acceleration of where is measured in feet per second per second. find the location of the object after sec. 73. show that if the speed of a particle traveling along a curve represented by a vector - valued function is constant, then the velocity function is always perpendicular to the acceleration function. 74. given and find 75. given find the velocity and the speed at any time. 76. find the velocity vector for the function 77. find the equation of the tangent line to the curve at | openstax_calculus_volume_3_-_web | [
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3. 2 • calculus of vector - valued functions 253 78. describe and sketch the curve represented by the vector - valued function 79. locate the highest point on the curve and give the value of the function at this point. the position vector for a particle is the graph is shown here : 80. find the velocity vector at any time. 81. find the speed of the particle at time sec. 82. find the acceleration at time sec. a particle travels along the path of a helix with the equation see the graph presented here : 254 3 • vector - valued functions access for free at openstax. org find the following : 83. velocity of the particle at any time 84. speed of the particle at any time 85. acceleration of the particle at any time 86. find the unit tangent vector for the helix. a particle travels along the path of an ellipse with the equation find the following : 87. velocity of the particle 88. speed of the particle at 89. acceleration of the particle at given the vector - valued function ( graph is shown here ), find the following : 90. velocity 91. speed 92. acceleration 93. find the minimum speed of a particle traveling along the curve given and find the following : 94. 95. 96. now, use the product rule for the derivative of the cross product of two vectors and show this result is the same as the answer for the preceding problem. 3. 2 • calculus of vector - valued functions 255 find the unit tangent vector t ( t ) for the following vector - valued functions. 97. the graph is shown here : 98. 99. evaluate the following integrals : 100. 101. where | openstax_calculus_volume_3_-_web | [
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3. 3 arc length and curvature learning objectives 3. 3. 1 determine the length of a particle ’ s path in space by using the arc - length function. 3. 3. 2 explain the meaning of the curvature of a curve in space and state its formula. 3. 3. 3 describe the meaning of the normal and binormal vectors of a curve in space. in this section, we study formulas related to curves in both two and three dimensions, and see how they are related to various properties of the same curve. for example, suppose a vector - valued function describes the motion of a particle in space. we would like to determine how far the particle has traveled over a given time interval, which can be described by the arc length of the path it follows. or, suppose that the vector - valued function describes a road we are building and we want to determine how sharply the road curves at a given point. this is described by the curvature of the function at that point. we explore each of these concepts in this section. arc length for vector functions we have seen how a vector - valued function describes a curve in either two or three dimensions. recall arc length of a parametric curve, which states that the formula for the arc length of a curve defined by the parametric functions is given by in a similar fashion, if we define a smooth curve using a vector - valued function where the arc length is given by the formula in three dimensions, if the vector - valued function is described by over the same interval the arc length is given by 256 3 • vector - valued functions access for free at openstax. org theorem 3. 4 arc - length formulas i. plane curve : given a smooth curve c defined by the function where t lies within the interval the arc length of c over the interval is ii. space curve : given a smooth curve c defined by the function where t lies within the interval the arc length of c over the interval is the two formulas are very similar ; they differ only in the fact that a space curve has three component functions instead of two. note that the formulas are defined for smooth curves : curves where the vector - valued function is differentiable with a non - zero derivative. the smoothness condition | openstax_calculus_volume_3_-_web | [
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over the interval is the two formulas are very similar ; they differ only in the fact that a space curve has three component functions instead of two. note that the formulas are defined for smooth curves : curves where the vector - valued function is differentiable with a non - zero derivative. the smoothness condition guarantees that the curve has no cusps ( or corners ) that could make the formula problematic. example 3. 9 finding the arc length calculate the arc length for each of the following vector - valued functions : a. b. solution a. using equation 3. 11, so b. using equation 3. 12, so here we can use a table integration formula ( 3. 11 ) ( 3. 12 ) 3. 3 • arc length and curvature 257 so we obtain | openstax_calculus_volume_3_-_web | [
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3. 9 calculate the arc length of the parameterized curve we now return to the helix introduced earlier in this chapter. a vector - valued function that describes a helix can be written in the form where r represents the radius of the helix, h represents the height ( distance between two consecutive turns ), and the helix completes n turns. let ’ s derive a formula for the arc length of this helix using equation 3. 12. first of all, therefore, this gives a formula for the length of a wire needed to form a helix with n turns that has radius r and height h. arc - length parameterization we now have a formula for the arc length of a curve defined by a vector - valued function. let ’ s take this one step further and examine what an arc - length function is. if a vector - valued function represents the position of a particle in space as a function of time, then the arc - length function measures how far that particle travels as a function of time. the formula for the arc - length function follows directly from the formula for arc length : if the curve is in two dimensions, then only two terms appear under the square root inside the integral. the reason for using the independent variable u is to distinguish between time and the variable of integration. since measures ( 3. 13 ) 258 3 • vector - valued functions access for free at openstax. org distance traveled as a function of time, measures the speed of the particle at any given time. since we have a formula for in equation 3. 13, we can differentiate both sides of the equation : if we assume that defines a smooth curve, then the arc length is always increasing, so for last, if is a curve on which for all t, then which means that t represents the arc length as long as theorem 3. 5 arc - length function let describe a smooth curve for then the arc - length function is given by furthermore, if for all then the parameter t represents the arc length from the starting point at a useful application of this theorem is to find an alternative parameterization of a given curve, called an arc - length parameterization. recall that any vector - valued function can be reparameterized via a change of variables. for example | openstax_calculus_volume_3_-_web | [
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then the parameter t represents the arc length from the starting point at a useful application of this theorem is to find an alternative parameterization of a given curve, called an arc - length parameterization. recall that any vector - valued function can be reparameterized via a change of variables. for example, if we have a function that parameterizes a circle of radius 3, we can change the parameter from t to obtaining a new parameterization the new parameterization still defines a circle of radius 3, but now we need only use the values to traverse the circle once. suppose that we find the arc - length function and are able to solve this function for t as a function of s. we can then reparameterize the original function by substituting the expression for t back into the vector - valued function is now written in terms of the parameter s. since the variable s represents the arc length, we call this an arc - length parameterization of the original function one advantage of finding the arc - length parameterization is that the distance traveled along the curve starting from is now equal to the parameter s. the arc - length parameterization also appears in the context of curvature ( which we examine later in this section ) and line integrals, which we study in the introduction to vector calculus. example 3. 10 finding an arc - length parameterization find the arc - length parameterization for each of the following curves : a. b. solution a. first we find the arc - length function using equation 3. 14 : ( 3. 14 ) 3. 3 • arc length and curvature 259 which gives the relationship between the arc length s and the parameter t as so, next we replace the variable t in the original function with the expression to obtain this is the arc - length parameterization of since the original restriction on t was given by the restriction on s becomes or b. the arc - length function is given by equation 3. 14 : therefore, the relationship between the arc length s and the parameter t is so substituting this into the original function yields this is an arc - length parameterization of the original restriction on the parameter was so the restriction on s is or | openstax_calculus_volume_3_-_web | [
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3. 10 find the arc - length function for the helix then, use the relationship between the arc length and the parameter t to find an arc - length parameterization of curvature an important topic related to arc length is curvature. the concept of curvature provides a way to measure how sharply a smooth curve turns. a circle has constant curvature. the smaller the radius of the circle, the greater the curvature. think of driving down a road. suppose the road lies on an arc of a large circle. in this case you would barely have to turn the wheel to stay on the road. now suppose the radius is smaller. in this case you would need to turn more sharply to stay on the road. in the case of a curve other than a circle, it is often useful first to inscribe a circle to the curve at a given point so that it is tangent to the curve at that point and “ hugs ” the curve as closely as possible in a neighborhood of the 260 3 • vector - valued functions access for free at openstax. org point ( figure 3. 6 ). the curvature of the graph at that point is then defined to be the same as the curvature of the inscribed circle. figure 3. 6 the graph represents the curvature of a function the sharper the turn in the graph, the greater the curvature, and the smaller the radius of the inscribed circle. definition let c be a smooth curve in the plane or in space given by where is the arc - length parameter. the curvature at s is media visit this website ( http : / / www. openstax. org / l / 20 _ spacecurve ) for more information about the curvature of a space curve. the formula in the definition of curvature is not very useful in terms of calculation. in particular, recall that represents the unit tangent vector to a given vector - valued function and the formula for is to use the formula for curvature, it is first necessary to express in terms of the arc - length parameter s, then find the unit tangent vector for the function then take the derivative of with respect to s. this is a tedious process. fortunately, there are equivalent formulas for curvature. theorem 3. 6 alternative formulas for curvature if c is a | openstax_calculus_volume_3_-_web | [
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necessary to express in terms of the arc - length parameter s, then find the unit tangent vector for the function then take the derivative of with respect to s. this is a tedious process. fortunately, there are equivalent formulas for curvature. theorem 3. 6 alternative formulas for curvature if c is a smooth curve given by then the curvature of c at t is given by if c is a three - dimensional curve, then the curvature can be given by the formula if c is the graph of a function and both and exist, then the curvature at point is given by proof the first formula follows directly from the chain rule : where s is the arc length along the curve c. dividing both sides by and taking the magnitude of both sides gives ( 3. 15 ) ( 3. 16 ) ( 3. 17 ) | openstax_calculus_volume_3_-_web | [
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3. 3 • arc length and curvature 261 since this gives the formula for the curvature of a curve c in terms of any parameterization of c : in the case of a three - dimensional curve, we start with the formulas and therefore, we can take the derivative of this function using the scalar product formula : using these last two equations we get since this reduces to since is parallel to and is orthogonal to it follows that and are orthogonal. this means that so now we solve this equation for and use the fact that then, we divide both sides by this gives this proves equation 3. 16. to prove equation 3. 17, we start with the assumption that curve c is defined by the function then, we can define using the previous formula for curvature : therefore, example 3. 11 finding curvature find the curvature for each of the following curves at the given point : a. 262 3 • vector - valued functions access for free at openstax. org b. solution a. this function describes a helix. the curvature of the helix at can be found by using equation 3. 15. first, calculate next, calculate last, apply equation 3. 15 : the curvature of this helix is constant at all points on the helix. b. this function describes a semicircle. 3. 3 • arc length and curvature 263 to find the curvature of this graph, we must use equation 3. 16. first, we calculate and then, we apply equation 3. 17 : the curvature of this circle is equal to the reciprocal of its radius. there is a minor issue with the absolute value in equation 3. 16 ; however, a closer look at the calculation reveals that the denominator is positive for any value of x. | openstax_calculus_volume_3_-_web | [
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3. 11 find the curvature of the curve defined by the function at the point the normal and binormal vectors we have seen that the derivative of a vector - valued function is a tangent vector to the curve defined by and the unit tangent vector can be calculated by dividing by its magnitude. when studying motion in three dimensions, two other vectors are useful in describing the motion of a particle along a path in space : the principal unit normal vector and the binormal vector. 264 3 • vector - valued functions access for free at openstax. org definition let c be a three - dimensional smooth curve represented by r over an open interval i. if then the principal unit normal vector at t is defined to be the binormal vector at t is defined as where is the unit tangent vector. note that, by definition, the binormal vector is orthogonal to both the unit tangent vector and the normal vector. furthermore, is always a unit vector. this can be shown using the formula for the magnitude of a cross product where is the angle between and since is the derivative of a unit vector, property ( vii ) of the derivative of a vector - valued function tells us that and are orthogonal to each other, so furthermore, they are both unit vectors, so their magnitude is 1. therefore, and is a unit vector. the principal unit normal vector can be challenging to calculate because the unit tangent vector involves a quotient, and this quotient often has a square root in the denominator. in the three - dimensional case, finding the cross product of the unit tangent vector and the unit normal vector can be even more cumbersome. fortunately, we have alternative formulas for finding these two vectors, and they are presented in motion in space. example 3. 12 finding the principal unit normal vector and binormal vector for each of the following vector - valued functions, find the principal unit normal vector. then, if possible, find the binormal vector. a. b. solution a. this function describes a circle. to find the principal unit normal vector, we first must find the unit tangent vector ( 3. 18 ) ( 3. 19 ) 3. 3 • arc length and curvature 265 next, we use | openstax_calculus_volume_3_-_web | [
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, find the binormal vector. a. b. solution a. this function describes a circle. to find the principal unit normal vector, we first must find the unit tangent vector ( 3. 18 ) ( 3. 19 ) 3. 3 • arc length and curvature 265 next, we use equation 3. 18 : notice that the unit tangent vector and the principal unit normal vector are orthogonal to each other for all values of t : furthermore, the principal unit normal vector points toward the center of the circle from every point on the circle. since defines a curve in two dimensions, we cannot calculate the binormal vector. b. this function looks like this : 266 3 • vector - valued functions access for free at openstax. org to find the principal unit normal vector, we first find the unit tangent vector next, we calculate and 3. 3 • arc length and curvature 267 therefore, according to equation 3. 18 : once again, the unit tangent vector and the principal unit normal vector are orthogonal to each other for all values of t : last, since represents a three - dimensional curve, we can calculate the binormal vector using equation 3. 17 : | openstax_calculus_volume_3_-_web | [
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3. 12 find the unit normal vector for the vector - valued function and evaluate it at for any smooth curve in three dimensions that is defined by a vector - valued function, we now have formulas for the unit tangent vector t, the unit normal vector n, and the binormal vector b. the unit normal vector and the binormal vector form a plane that is perpendicular to the curve at any point on the curve, called the normal plane. in addition, these three vectors form a frame of reference in three - dimensional space called the frenet frame of reference ( also called the tnb frame ) ( figure 3. 7 ). last, the plane determined by the vectors t and n forms the osculating plane of c at any point p on the curve. 268 3 • vector - valued functions access for free at openstax. org figure 3. 7 this figure depicts a frenet frame of reference. at every point p on a three - dimensional curve, the unit tangent, unit normal, and binormal vectors form a three - dimensional frame of reference. suppose we form a circle in the osculating plane of c at point p on the curve. assume that the circle has the same curvature as the curve does at point p and let the circle have radius r. then, the curvature of the circle is given by we call r the radius of curvature of the curve, and it is equal to the reciprocal of the curvature. if this circle lies on the concave side of the curve and is tangent to the curve at point p, then this circle is called the osculating circle of c at p, as shown in the following figure. figure 3. 8 in this osculating circle, the circle is tangent to curve c at point p and shares the same curvature. media for more information on osculating circles, see this demonstration ( http : / / www. openstax. org / l / 20 _ osculcircle1 ) on curvature and torsion, this article ( http : / / www. openstax. org / l / 20 _ osculcircle3 ) on osculating circles, and this discussion ( http : / / www | openstax_calculus_volume_3_-_web | [
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org / l / 20 _ osculcircle1 ) on curvature and torsion, this article ( http : / / www. openstax. org / l / 20 _ osculcircle3 ) on osculating circles, and this discussion ( http : / / www. openstax. org / l / 20 _ osculcircle2 ) of serret formulas. to find the equation of an osculating circle in two dimensions, we need find only the center and radius of the circle. example 3. 13 finding the equation of an osculating circle find the equation of the osculating circle of the helix defined by the function at solution figure 3. 9 shows the graph of | openstax_calculus_volume_3_-_web | [
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3. 3 • arc length and curvature 269 figure 3. 9 we want to find the osculating circle of this graph at the point where first, let ’ s calculate the curvature at this gives therefore, the radius of the osculating circle is given by next, we then calculate the coordinates of the center of the circle. when the slope of the tangent line is zero. therefore, the center of the osculating circle is directly above the point on the graph with coordinates the center is located at the formula for a circle with radius r and center is given by therefore, the equation of the osculating circle is the graph and its osculating circle appears in the following graph. figure 3. 10 the osculating circle has radius 3. 13 find the equation of the osculating circle of the curve defined by the vector - valued function at 270 3 • vector - valued functions access for free at openstax. org section 3. 3 exercises find the arc length of the curve on the given interval. 102. this portion of the graph is shown here : 103. 104. this portion of the graph is shown here : 105. 106. over the interval here is the portion of the graph on the indicated interval : 107. find the length of one turn of the helix given by 108. find the arc length of the vector - valued function over | openstax_calculus_volume_3_-_web | [
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3. 3 • arc length and curvature 271 109. a particle travels once around a circle with the equation of motion find the distance traveled around the circle by the particle. 110. set up an integral to find the circumference of the ellipse with the equation 111. find the length of the curve over the interval the graph is shown here : 112. find the length of the curve for 113. the position function for a particle is find the unit tangent vector and the unit normal vector at 114. given find the binormal vector 115. given determine the tangent vector 116. given determine the unit tangent vector evaluated at 117. given find the unit normal vector evaluated at 118. given find the unit binormal vector evaluated at 119. given find the unit tangent vector the graph is shown here : 120. find the unit tangent vector and unit normal vector at for the plane curve the graph is shown here : 121. find the unit tangent vector for 122. find the principal normal vector to the curve at the point determined by 123. find for the curve 124. find for the curve 125. find the unit normal vector for 126. find the unit tangent vector for 272 3 • vector - valued functions access for free at openstax. org 127. find the arc - length function for the line segment given by write r as a parameter of s. 128. parameterize the helix using the arc - length parameter s, from 129. parameterize the curve using the arc - length parameter s, at the point at which for 130. find the curvature of the curve at ( note : the graph is an ellipse. ) 131. find the x - coordinate at which the curvature of the curve is a maximum value. 132. find the curvature of the curve does the curvature depend upon the parameter t? 133. find the curvature for the curve at the point 134. find the curvature for the curve at the point 135. find the curvature of the curve the graph is shown here : 136. find the curvature of 137. find the curvature of at point 138. at what point does the curve have maximum curvature? 139. what happens to the curvature as ∞for the curve 140. find the point of maximum curvature on the | openstax_calculus_volume_3_-_web | [
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find the curvature of the curve the graph is shown here : 136. find the curvature of 137. find the curvature of at point 138. at what point does the curve have maximum curvature? 139. what happens to the curvature as ∞for the curve 140. find the point of maximum curvature on the curve 141. find the equations of the normal plane and the osculating plane of the curve at point 142. find equations of the osculating circles of the ellipse at the points and 143. find the equation for the osculating plane at point on the curve 144. find the radius of curvature of at the point | openstax_calculus_volume_3_-_web | [
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3. 3 • arc length and curvature 273 145. find the curvature at each point on the hyperbola 146. calculate the curvature of the circular helix 147. find the radius of curvature of at point 148. find the radius of curvature of the hyperbola at point a particle moves along the plane curve c described by solve the following problems. 149. find the length of the curve over the interval 150. find the curvature of the plane curve at 151. describe the curvature as t increases from to the surface of a large cup is formed by revolving the graph of the function from to about the y - axis ( measured in centimeters ). 152. [ t ] use technology to graph the surface. 153. find the curvature of the generating curve as a function of x. 154. [ t ] use technology to graph the curvature function. | openstax_calculus_volume_3_-_web | [
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3. 4 motion in space learning objectives 3. 4. 1 describe the velocity and acceleration vectors of a particle moving in space. 3. 4. 2 explain the tangential and normal components of acceleration. 3. 4. 3 state kepler ’ s laws of planetary motion. we have now seen how to describe curves in the plane and in space, and how to determine their properties, such as arc length and curvature. all of this leads to the main goal of this chapter, which is the description of motion along plane curves and space curves. we now have all the tools we need ; in this section, we put these ideas together and look at how to use them. motion vectors in the plane and in space our starting point is using vector - valued functions to represent the position of an object as a function of time. all of the following material can be applied either to curves in the plane or to space curves. for example, when we look at the orbit of the planets, the curves defining these orbits all lie in a plane because they are elliptical. however, a particle traveling along a helix moves on a curve in three dimensions. definition let be a twice - differentiable vector - valued function of the parameter t that represents the position of an object as a function of time. the velocity vector of the object is given by the acceleration vector is defined to be the speed is defined to be since can be in either two or three dimensions, these vector - valued functions can have either two or three components. in two dimensions, we define and in three dimensions then the velocity, acceleration, and speed can be written as shown in the following table. ( 3. 20 ) ( 3. 21 ) ( 3. 22 ) 274 3 • vector - valued functions access for free at openstax. org quantity two dimensions three dimensions position velocity acceleration speed table 3. 4 formulas for position, velocity, acceleration, and speed example 3. 14 studying motion along a parabola a particle moves in a parabolic path defined by the vector - valued function where t measures time in seconds. a. find the velocity, acceleration, and speed as functions of time. b. sketch the curve along with the velocity vector at time solution a. we use equation 3 | openstax_calculus_volume_3_-_web | [
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along a parabola a particle moves in a parabolic path defined by the vector - valued function where t measures time in seconds. a. find the velocity, acceleration, and speed as functions of time. b. sketch the curve along with the velocity vector at time solution a. we use equation 3. 20, equation 3. 21, and equation 3. 22 : b. the graph of is a portion of a parabola ( figure 3. 11 ). the velocity vector at is and the acceleration vector at is notice that the velocity vector is tangent to the path, as is always the case. 3. 4 • motion in space 275 figure 3. 11 this graph depicts the velocity vector at time for a particle moving in a parabolic path. | openstax_calculus_volume_3_-_web | [
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3. 14 a particle moves in a path defined by the vector - valued function where t measures time in seconds and where distance is measured in feet. find the velocity, acceleration, and speed as functions of time. to gain a better understanding of the velocity and acceleration vectors, imagine you are driving along a curvy road. if you do not turn the steering wheel, you would continue in a straight line and run off the road. the speed at which you are traveling when you run off the road, coupled with the direction, gives a vector representing your velocity, as illustrated in the following figure. figure 3. 12 at each point along a road traveled by a car, the velocity vector of the car is tangent to the path traveled by the car. however, the fact that you must turn the steering wheel to stay on the road indicates that your velocity is always changing ( even if your speed is not ) because your direction is constantly changing to keep you on the road. as you turn to the right, your acceleration vector also points to the right. as you turn to the left, your acceleration vector points to the left. this indicates that your velocity and acceleration vectors are constantly changing, regardless of whether your actual speed varies ( figure 3. 13 ). 276 3 • vector - valued functions access for free at openstax. org figure 3. 13 the dashed line represents the trajectory of an object ( a car, for example ). the acceleration vector points toward the inside of the turn at all times. components of the acceleration vector we can combine some of the concepts discussed in arc length and curvature with the acceleration vector to gain a deeper understanding of how this vector relates to motion in the plane and in space. recall that the unit tangent vector t and the unit normal vector n form an osculating plane at any point p on the curve defined by a vector - valued function the following theorem shows that the acceleration vector lies in the osculating plane and can be written as a linear combination of the unit tangent and the unit normal vectors. theorem 3. 7 the plane of the acceleration vector the acceleration vector of an object moving along a curve traced out by a twice - differentiable function lies in the plane formed by the unit tangent vector | openstax_calculus_volume_3_-_web | [
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##culating plane and can be written as a linear combination of the unit tangent and the unit normal vectors. theorem 3. 7 the plane of the acceleration vector the acceleration vector of an object moving along a curve traced out by a twice - differentiable function lies in the plane formed by the unit tangent vector and the principal unit normal vector to c. furthermore, here, is the speed of the object and is the curvature of c traced out by proof because and we have now we differentiate this equation : since we know so a formula for curvature is so this gives the coefficients of and are referred to as the tangential component of acceleration and the normal component of acceleration, respectively. we write to denote the tangential component and to denote the normal component. | openstax_calculus_volume_3_-_web | [
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3. 4 • motion in space 277 theorem 3. 8 tangential and normal components of acceleration let be a vector - valued function that denotes the position of an object as a function of time. then is the acceleration vector. the tangential and normal components of acceleration and are given by the formulas and these components are related by the formula here is the unit tangent vector to the curve defined by and is the unit normal vector to the curve defined by the normal component of acceleration is also called the centripetal component of acceleration or sometimes the radial component of acceleration. to understand centripetal acceleration, suppose you are traveling in a car on a circular track at a constant speed. then, as we saw earlier, the acceleration vector points toward the center of the track at all times. as a rider in the car, you feel a pull toward the outside of the track because you are constantly turning. this sensation acts in the opposite direction of centripetal acceleration. the same holds true for noncircular paths. the reason is that your body tends to travel in a straight line and resists the force resulting from acceleration that push it toward the side. note that at point b in figure 3. 14 the acceleration vector is pointing backward. this is because the car is decelerating as it goes into the curve. figure 3. 14 the tangential and normal components of acceleration can be used to describe the acceleration vector. the tangential and normal unit vectors at any given point on the curve provide a frame of reference at that point. the tangential and normal components of acceleration are the projections of the acceleration vector onto t and n, respectively. example 3. 15 finding components of acceleration a particle moves in a path defined by the vector - valued function where t measures time in seconds and distance is measured in feet. a. find and as functions of t. b. find and at time solution a. let ’ s start with equation 3. 23 : ( 3. 23 ) ( 3. 24 ) ( 3. 25 ) 278 3 • vector - valued functions access for free at openstax. org then we apply equation 3. 24 : b. we must evaluate each of the answers from part a. at the units | openstax_calculus_volume_3_-_web | [
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equation 3. 23 : ( 3. 23 ) ( 3. 24 ) ( 3. 25 ) 278 3 • vector - valued functions access for free at openstax. org then we apply equation 3. 24 : b. we must evaluate each of the answers from part a. at the units of acceleration are feet per second squared, as are the units of the normal and tangential components of acceleration. | openstax_calculus_volume_3_-_web | [
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3. 15 an object moves in a path defined by the vector - valued function where t measures time in seconds. a. find and as functions of t. b. find and at time projectile motion now let ’ s look at an application of vector functions. in particular, let ’ s consider the effect of gravity on the motion of an object as it travels through the air, and how it determines the resulting trajectory of that object. in the following, we ignore the effect of air resistance. this situation, with an object moving with an initial velocity but with no forces acting on it other than gravity, is known as projectile motion. it describes the motion of objects from golf balls to baseballs, and from arrows to cannonballs. first we need to choose a coordinate system. if we are standing at the origin of this coordinate system, then we choose the positive y - axis to be up, the negative y - axis to be down, and the positive x - axis to be forward ( i. e., away from the thrower of the object ). the effect of gravity is in a downward direction, so newton ’ s second law tells us that the force on | openstax_calculus_volume_3_-_web | [
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3. 4 • motion in space 279 the object resulting from gravity is equal to the mass of the object times the acceleration resulting from to gravity, or where represents the force from gravity and g represents the acceleration resulting from gravity at earth ’ s surface. the value of g in the english system of measurement is approximately 32 ft / sec2 and it is approximately 9. 8 m / sec2 in the metric system. this is the only force acting on the object. since gravity acts in a downward direction, we can write the force resulting from gravity in the form as shown in the following figure. figure 3. 15 an object is falling under the influence of gravity. media visit this website ( http : / / www. openstax. org / l / 20 _ projectile ) for a video showing projectile motion. newton ’ s second law also tells us that where a represents the acceleration vector of the object. this force must be equal to the force of gravity at all times, so we therefore know that now we use the fact that the acceleration vector is the first derivative of the velocity vector. therefore, we can rewrite the last equation in the form by taking the antiderivative of each side of this equation we obtain for some constant vector to determine the value of this vector, we can use the velocity of the object at a fixed time, say at time we call this velocity the initial velocity : therefore, and this gives the velocity vector as next we use the fact that velocity is the derivative of position this gives the equation taking the antiderivative of both sides of this equation leads to with another unknown constant vector to determine the value of we can use the position of the object at a given time, say at time we call this position the initial position : therefore, and this gives the position of the object at any time as 280 3 • vector - valued functions access for free at openstax. org let ’ s take a closer look at the initial velocity and initial position. in particular, suppose the object is thrown upward from the origin at an angle to the horizontal, with initial speed how can we modify the previous result to reflect this scenario? first, we can assume it is thrown from the origin. if not, then | openstax_calculus_volume_3_-_web | [
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