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Get areas and volumes from lengths. | https://openstax.org/books/university-physics-volume-1/pages/1-summary |
Get masses from volumes and densities. | https://openstax.org/books/university-physics-volume-1/pages/1-summary |
If all else fails, bound it. | https://openstax.org/books/university-physics-volume-1/pages/1-summary |
One âsig. fig.â is fine. | https://openstax.org/books/university-physics-volume-1/pages/1-summary |
Ask yourself: Does this make any sense? | https://openstax.org/books/university-physics-volume-1/pages/1-summary |
Accuracy of a measured value refers to how close a measurement is to an accepted reference value. The discrepancy in a measurement is the amount by which the measurement result differs from this value. | https://openstax.org/books/university-physics-volume-1/pages/1-summary |
Precision of measured values refers to how close the agreement is between repeated measurements. The uncertainty of a measurement is a quantification of this. | https://openstax.org/books/university-physics-volume-1/pages/1-summary |
The precision of a measuring tool is related to the size of its measurement increments. The smaller the measurement increment, the more precise the tool. | https://openstax.org/books/university-physics-volume-1/pages/1-summary |
Significant figures express the precision of a measuring tool. | https://openstax.org/books/university-physics-volume-1/pages/1-summary |
When multiplying or dividing measured values, the final answer can contain only as many significant figures as the value with the least number of significant figures. | https://openstax.org/books/university-physics-volume-1/pages/1-summary |
When adding or subtracting measured values, the final answer cannot contain more decimal places than the least-precise value. | https://openstax.org/books/university-physics-volume-1/pages/1-summary |
The three stages of the process for solving physics problems used in this book are as follows: | https://openstax.org/books/university-physics-volume-1/pages/1-summary |
Strategy: Determine which physical principles are involved and develop a strategy for using them to solve the problem. | https://openstax.org/books/university-physics-volume-1/pages/1-summary |
Solution: Do the math necessary to obtain a numerical solution complete with units. | https://openstax.org/books/university-physics-volume-1/pages/1-summary |
Significance: Check the solution to make sure it makes sense (correct units, reasonable magnitude and sign) and assess its significance. | https://openstax.org/books/university-physics-volume-1/pages/1-summary |
B â = α A â B â = α A â | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
B = | α | A B = | α | A | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
D â A D = D â A C + D â C D D â A D = D â A C + D â C D | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
A â + B â = B â + A â A â + B â = B â + A â | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
( A â + B â ) + C â = A â + ( B â + C â ) ( A â + B â ) + C â = A â + ( B â + C â ) | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
α 1 A â + α 2 A â = ( α 1 + α 2 ) A â α 1 A â + α 2 A â = ( α 1 + α 2 ) A â | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
A â = A x i ^ + A y j ^ A â = A x i ^ + A y j ^ | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
{ A x = x e â x b A y = y e â y b { A x = x e â x b A y = y e â y b | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
A = A x 2 + A y 2 A = A x 2 + A y 2 | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
θ A = tan â1 ( A y A x ) θ A = tan â1 ( A y A x ) | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
{ A x = A cos θ A A y = A sin θ A { A x = A cos θ A A y = A sin θ A | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
{ x = r cos Ï y = r sin Ï { x = r cos Ï y = r sin Ï | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
A â = A x i ^ + A y j ^ + A z k ^ A â = A x i ^ + A y j ^ + A z k ^ | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
A z = z e â z b A z = z e â z b | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
A = A x 2 + A y 2 + A z 2 A = A x 2 + A y 2 + A z 2 | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
α ( A â + B â ) = α A â + α B â α ( A â + B â ) = α A â + α B â | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
â A â = â A x i ^ â A y j ^ â A z k ^ â A â = â A x i ^ â A y j ^ â A z k ^ | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
A â = B â â { A x = B x A y = B y A z = B z A â = B â â { A x = B x A y = B y A z = B z | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
{ F R x = â k = 1 N F k x = F 1 x + F 2 x + ⦠+ F N x F R y = â k = 1 N F k y = F 1 y + F 2 y + ⦠+ F N y F R z = â k = 1 N F k z = F 1 z + F 2 z + ⦠+ F N z { F R x = â k = 1 N F k x = F 1 x + F 2 x + ⦠+ F N x F R y = â k = 1 N F k y = F 1 y + F 2 y + ⦠+ F N y F R z = â k = 1 N F k z = F 1 z + ... | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
V ^ = V â V V ^ = V â V | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
A â · B â = A B cos Ï A â · B â = A B cos Ï | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
A â · B â = B â · A â A â · B â = B â · A â | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
A â · ( B â + C â ) = A â · B â + A â · C â A â · ( B â + C â ) = A â · B â + A â · C â | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
A â · B â = A x B x + A y B y + A z B z A â · B â = A x B x + A y B y + A z B z | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
cos Ï = A â · B â A B cos Ï = A â · B â A B | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
i ^ · j ^ = j ^ · k ^ = k ^ · i ^ = 0 i ^ · j ^ = j ^ · k ^ = k ^ · i ^ = 0 | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
| A â Ã B â | = A B sin Ï | A â Ã B â | = A B sin Ï | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
A â Ã B â = â B â Ã A â A â Ã B â = â B â Ã A â | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
A â Ã ( B â + C â ) = A â Ã B â + A â Ã C â A â Ã ( B â + C â ) = A â Ã B â + A â Ã C â | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
{ i ^ Ã j ^ = + k ^ , j ^ Ã k ^ = + i ^ , k ^ Ã i ^ = + j ^ . { i ^ Ã j ^ = + k ^ , j ^ Ã k ^ = + i ^ , k ^ Ã i ^ = + j ^ . | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
A â Ã B â = ( A y B z â A z B y ) i ^ + ( A z B x â A x B z ) j ^ + ( A x B y â A y B x ) k ^ A â Ã B â = ( A y B z â A z B y ) i ^ + ( A z B x â A x B z ) j ^ + ( A x B y â A y B x ) k ^ | https://openstax.org/books/university-physics-volume-1/pages/2-key-equations |
anticommutative property : change in the order of operation introduces the minus sign | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
antiparallel vectors : two vectors with directions that differ by180°180° | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
associative : terms can be grouped in any fashion | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
commutative : operations can be performed in any order | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
component form of a vector : a vector written as the vector sum of its components in terms of unit vectors | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
corkscrew right-hand rule : a rule used to determine the direction of the vector product | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
cross product : the result of the vector multiplication of vectors is a vector called a cross product; also called a vector product | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
difference of two vectors : vector sum of the first vector with the vector antiparallel to the second | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
direction angle : in a plane, an angle between the positive direction of thex-axis and the vector, measured counterclockwise from the axis to the vector | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
displacement : change in position | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
distributive : multiplication can be distributed over terms in summation | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
dot product : the result of the scalar multiplication of two vectors is a scalar called a dot product; also called a scalar product | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
equal vectors : two vectors are equal if and only if all their corresponding components are equal; alternately, two parallel vectors of equal magnitudes | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
magnitude : length of a vector | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
null vector : a vector with all its components equal to zero | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
orthogonal vectors : two vectors with directions that differ by exactly90°90°, synonymous with perpendicular vectors | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
parallel vectors : two vectors with exactly the same direction angles | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
parallelogram rule : geometric construction of the vector sum in a plane | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
polar coordinate system : an orthogonal coordinate system where location in a plane is given by polar coordinates | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
polar coordinates : a radial coordinate and an angle | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
radial coordinate : distance to the origin in a polar coordinate system | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
resultant vector : vector sum of two (or more) vectors | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
scalar : a number, synonymous with a scalar quantity in physics | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
scalar component : a number that multiplies a unit vector in a vector component of a vector | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
scalar equation : equation in which the left-hand and right-hand sides are numbers | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
scalar product : the result of the scalar multiplication of two vectors is a scalar called a scalar product; also called a dot product | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
scalar quantity : quantity that can be specified completely by a single number with an appropriate physical unit | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
tail-to-head geometric construction : geometric construction for drawing the resultant vector of many vectors | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
unit vector : vector of a unit magnitude that specifies direction; has no physical unit | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
unit vectors of the axes : unit vectors that define orthogonal directions in a plane or in space | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
vector : mathematical object with magnitude and direction | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
vector components : orthogonal components of a vector; a vector is the vector sum of its vector components. | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
vector equation : equation in which the left-hand and right-hand sides are vectors | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
vector product : the result of the vector multiplication of vectors is a vector called a vector product; also called a cross product | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
vector quantity : physical quantity described by a mathematical vectorâthat is, by specifying both its magnitude and its direction; synonymous with a vector in physics | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
vector sum : resultant of the combination of two (or more) vectors | https://openstax.org/books/university-physics-volume-1/pages/2-key-terms |
A vector quantity is any quantity that has magnitude and direction, such as displacement or velocity. Vector quantities are represented by mathematical objects called vectors. | https://openstax.org/books/university-physics-volume-1/pages/2-summary |
Geometrically, vectors are represented by arrows, with the end marked by an arrowhead. The length of the vector is its magnitude, which is a positive scalar. On a plane, the direction of a vector is given by the angle the vector makes with a reference direction, often an angle with the horizontal. The direction angle o... | https://openstax.org/books/university-physics-volume-1/pages/2-summary |
Two vectors are equal if and only if they have the same magnitudes and directions. Parallel vectors have the same direction angles but may have different magnitudes. Antiparallel vectors have direction angles that differ by180°180°. Orthogonal vectors have direction angles that differ by90°90°. | https://openstax.org/books/university-physics-volume-1/pages/2-summary |
When a vector is multiplied by a scalar, the result is another vector of a different length than the length of the original vector. Multiplication by a positive scalar does not change the original direction; only the magnitude is affected. Multiplication by a negative scalar reverses the original direction. The resulti... | https://openstax.org/books/university-physics-volume-1/pages/2-summary |
Two or more vectors can be added to form another vector. The vector sum is called the resultant vector. We can add vectors to vectors or scalars to scalars, but we cannot add scalars to vectors. Vector addition is commutative and associative. | https://openstax.org/books/university-physics-volume-1/pages/2-summary |
To construct a resultant vector of two vectors in a plane geometrically, we use the parallelogram rule. To construct a resultant vector of many vectors in a plane geometrically, we use the tail-to-head method. | https://openstax.org/books/university-physics-volume-1/pages/2-summary |
Vectors are described in terms of their components in a coordinate system. In two dimensions (in a plane), vectors have two components. In three dimensions (in space), vectors have three components. | https://openstax.org/books/university-physics-volume-1/pages/2-summary |
A vector component of a vector is its part in an axis direction. The vector component is the product of the unit vector of an axis with its scalar component along this axis. A vector is the resultant of its vector components. | https://openstax.org/books/university-physics-volume-1/pages/2-summary |
Scalar components of a vector are differences of coordinates, where coordinates of the origin are subtracted from end point coordinates of a vector. In a rectangular system, the magnitude of a vector is the square root of the sum of the squares of its components. | https://openstax.org/books/university-physics-volume-1/pages/2-summary |
In a plane, the direction of a vector is given by an angle the vector has with the positivex-axis. This direction angle is measured counterclockwise. The scalarx-component of a vector can be expressed as the product of its magnitude with the cosine of its direction angle, and the scalary-component can be expressed as t... | https://openstax.org/books/university-physics-volume-1/pages/2-summary |
In a plane, there are two equivalent coordinate systems. The Cartesian coordinate system is defined by unit vectorsi^i^andj^j^along thex-axis and they-axis, respectively. The polar coordinate system is defined by the radial unit vectorr^r^, which gives the direction from the origin, and a unit vectort^t^, which is perp... | https://openstax.org/books/university-physics-volume-1/pages/2-summary |
Analytical methods of vector algebra allow us to find resultants of sums or differences of vectors without having to draw them. Analytical methods of vector addition are exact, contrary to graphical methods, which are approximate. | https://openstax.org/books/university-physics-volume-1/pages/2-summary |
Analytical methods of vector algebra are used routinely in mechanics, electricity, and magnetism. They are important mathematical tools of physics. | https://openstax.org/books/university-physics-volume-1/pages/2-summary |
There are two kinds of multiplication for vectors. One kind of multiplication is the scalar product, also known as the dot product. The other kind of multiplication is the vector product, also known as the cross product. The scalar product of vectors is a number (scalar). The vector product of vectors is a vector. | https://openstax.org/books/university-physics-volume-1/pages/2-summary |
Both kinds of multiplication have the distributive property, but only the scalar product has the commutative property. The vector product has the anticommutative property, which means that when we change the order in which two vectors are multiplied, the result acquires a minus sign. | https://openstax.org/books/university-physics-volume-1/pages/2-summary |
The scalar product of two vectors is obtained by multiplying their magnitudes with the cosine of the angle between them. The scalar product of orthogonal vectors vanishes; the scalar product of antiparallel vectors is negative. | https://openstax.org/books/university-physics-volume-1/pages/2-summary |
The vector product of two vectors is a vector perpendicular to both of them. Its magnitude is obtained by multiplying their magnitudes by the sine of the angle between them. The direction of the vector product can be determined by the corkscrew right-hand rule. The vector product of two either parallel or antiparallel ... | https://openstax.org/books/university-physics-volume-1/pages/2-summary |
The scalar product of vectors is used to find angles between vectors and in the definitions of derived scalar physical quantities such as work or energy. | https://openstax.org/books/university-physics-volume-1/pages/2-summary |
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