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If the position vector is the same for two vectors, they are equal. SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-concepts |
Vectors are defined by their magnitude and direction. SeeExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-concepts |
If two vectors have the same magnitude and direction, they are equal. SeeExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-concepts |
Vector addition and subtraction result in a new vector found by adding or subtracting corresponding elements. SeeExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-concepts |
Scalar multiplication is multiplying a vector by a constant. Only the magnitude changes; the direction stays the same. SeeExample 6andExample 7. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-concepts |
Vectors are comprised of two components: the horizontal component along the positivex-axis, and the vertical component along the positivey-axis. SeeExample 8. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-concepts |
The unit vector in the same direction of any nonzero vector is found by dividing the vector by its magnitude. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-concepts |
The magnitude of a vector in the rectangular coordinate system is|v|=a2+b2.|v|=a2+b2.SeeExample 9. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-concepts |
In the rectangular coordinate system, unit vectors may be represented in terms ofiiandjjwhereiirepresents the horizontal component andjjrepresents the vertical component. Then,v= ai+ bjâ is a scalar multiple ofvvby real numbersaandb.aandb.SeeExample 10andExample 11. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-concepts |
Adding and subtracting vectors in terms ofiandjconsists of adding or subtracting corresponding coefficients ofiand corresponding coefficients ofj. SeeExample 12. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-concepts |
A vectorv=ai+bjis written in terms of magnitude and direction asv=|v|cosθi+|v|sinθj.v=|v|cosθi+|v|sinθj.SeeExample 13. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-concepts |
The dot product of two vectors is the product of theiiterms plus the product of thejjterms. SeeExample 14. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-concepts |
We can use the dot product to find the angle between two vectors.Example 15andExample 16. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-concepts |
Dot products are useful for many types of physics applications. SeeExample 17. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-concepts |
sin α a = sin β b = sin γ c a sin α = b sin β = c sin γ sin α a = sin β b = sin γ c a sin α = b sin β = c sin γ | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-equations |
Area = 1 2 b c sin α = 1 2 a c sin β = 1 2 a b sin γ Area = 1 2 b c sin α = 1 2 a c sin β = 1 2 a b sin γ | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-equations |
a 2 = b 2 + c 2 â 2 b c cos α b 2 = a 2 + c 2 â 2 a c cos β c 2 = a 2 + b 2 â 2 a b c o s γ a 2 = b 2 + c 2 â 2 b c cos α b 2 = a 2 + c 2 â 2 a c cos β c 2 = a 2 + b 2 â 2 a b c o s γ | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-equations |
Area = s ( s â a ) ( s â b ) ( s â c ) where s = ( a + b + c ) 2 Area = s ( s â a ) ( s â b ) ( s â c ) where s = ( a + b + c ) 2 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-equations |
cos θ = x r â x = r cos θ sin θ = y r â y = r sin θ r 2 = x 2 + y 2 tan θ = y x cos θ = x r â x = r cos θ sin θ = y r â y = r sin θ r 2 = x 2 + y 2 tan θ = y x | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-equations |
altitude : a perpendicular line from one vertex of a triangle to the opposite side, or in the case of an obtuse triangle, to the line containing the opposite side, forming two right triangles | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
ambiguous case : a scenario in which more than one triangle is a valid solution for a given oblique SSA triangle | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
Archimedesâ spiral : a polar curve given byr=θ.r=θ.When multiplied by a constant, the equation appears asr=aθ.r=aθ.Asr=θ,r=θ,the curve continues to widen in a spiral path over the domain. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
argument : the angle associated with a complex number; the angle between the line from the origin to the point and the positive real axis | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
cardioid : a member of the limaçon family of curves, named for its resemblance to a heart; its equation is given asr=a±bcosθr=a±bcosθandr=a±bsinθ,r=a±bsinθ,whereab=1ab=1 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
convex limaÒ«on : a type of one-loop limaçon represented byr=a±bcosθr=a±bcosθandr=a±bsinθr=a±bsinθsuch thatabâ¥2abâ¥2 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
De Moivreâs Theorem : formula used to find thenthnthpower ornth roots of a complex number; states that, for a positive integern,znn,znis found by raising the modulus to thenthnthpower and multiplying the angles bynn | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
dimpled limaҫon : a type of one-loop limaçon represented byr=a±bcosθr=a±bcosθandr=a±bsinθr=a±bsinθsuch that1<ab<21<ab<2 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
dot product : given two vectors, the sum of the product of the horizontal components and the product of the vertical components | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
Generalized Pythagorean Theorem : an extension of the Law of Cosines; relates the sides of an oblique triangle and is used for SAS and SSS triangles | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
initial point : the origin of a vector | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
inner-loop limaçon : a polar curve similar to the cardioid, but with an inner loop; passes through the pole twice; represented byr=a±bcosθr=a±bcosθandr=a±bsinθr=a±bsinθwherea<ba<b | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
Law of Cosines : states that the square of any side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of the other two sides and the cosine of the included angle | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
Law of Sines : states that the ratio of the measurement of one angle of a triangle to the length of its opposite side is equal to the remaining two ratios of angle measure to opposite side; any pair of proportions may be used to solve for a missing angle or side | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
lemniscate : a polar curve resembling a figure 8 and given by the equationr2=a2cos2θr2=a2cos2θandr2=a2sin2θ,r2=a2sin2θ,aâ0aâ0 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
magnitude : the length of a vector; may represent a quantity such as speed, and is calculated using the Pythagorean Theorem | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
modulus : the absolute value of a complex number, or the distance from the origin to the point(x,y);(x,y);also called the amplitude | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
oblique triangle : any triangle that is not a right triangle | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
one-loop limaҫon : a polar curve represented byr=a±bcosθr=a±bcosθandr=a±bsinθr=a±bsinθsuch thata>0,b>0,a>0,b>0,andab>1;ab>1;may be dimpled or convex; does not pass through the pole | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
parameter : a variable, often representing time, upon whichxxandyyare both dependent | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
polar axis : on the polar grid, the equivalent of the positivex-axis on the rectangular grid | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
polar coordinates : on the polar grid, the coordinates of a point labeled(r,θ),(r,θ),whereθθindicates the angle of rotation from the polar axis andrrrepresents the radius, or the distance of the point from the pole in the direction ofθθ | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
polar equation : an equation describing a curve on the polar grid. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
polar form of a complex number : a complex number expressed in terms of an angleθθand its distance from the originr;r;can be found by using conversion formulasx=rcosθ,y=rsinθ,x=rcosθ,y=rsinθ,andr=x2+y2r=x2+y2 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
pole : the origin of the polar grid | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
resultant : a vector that results from addition or subtraction of two vectors, or from scalar multiplication | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
rose curve : a polar equation resembling a flower, given by the equationsr=acosnθr=acosnθandr=asinnθ;r=asinnθ;whennnis even there are2n2npetals, and the curve is highly symmetrical; whennnis odd there arennpetals. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
scalar : a quantity associated with magnitude but not direction; a constant | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
scalar multiplication : the product of a constant and each component of a vector | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
standard position : the placement of a vector with the initial point at(0,0)(0,0)and the terminal point(a,b),(a,b),represented by the change in thex-coordinates and the change in they-coordinates of the original vector | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
terminal point : the end point of a vector, usually represented by an arrow indicating its direction | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
unit vector : a vector that begins at the origin and has magnitude of 1; the horizontal unit vector runs along thex-axis and is defined asv1=â©1,0âªv1=â©1,0âªthe vertical unit vector runs along they-axis and is defined asv2=â©0,1âª.v2=â©0,1âª. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
vector : a quantity associated with both magnitude and direction, represented as a directed line segment with a starting point (initial point) and an end point (terminal point) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
vector addition : the sum of two vectors, found by adding corresponding components | https://openstax.org/books/algebra-and-trigonometry-2e/pages/10-key-terms |
A system of linear equations consists of two or more equations made up of two or more variables such that all equations in the system are considered simultaneously. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
Systems of equations are classified as independent with one solution, dependent with an infinite number of solutions, or inconsistent with no solution. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
One method of solving a system of linear equations in two variables is by graphing. In this method, we graph the equations on the same set of axes. SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
Another method of solving a system of linear equations is by substitution. In this method, we solve for one variable in one equation and substitute the result into the second equation. SeeExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
A third method of solving a system of linear equations is by addition, in which we can eliminate a variable by adding opposite coefficients of corresponding variables. SeeExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
It is often necessary to multiply one or both equations by a constant to facilitate elimination of a variable when adding the two equations together. SeeExample 5,Example 6, andExample 7. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
Either method of solving a system of equations results in a false statement for inconsistent systems because they are made up of parallel lines that never intersect. SeeExample 8. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
The solution to a system of dependent equations will always be true because both equations describe the same line. SeeExample 9. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
Systems of equations can be used to solve real-world problems that involve more than one variable, such as those relating to revenue, cost, and profit. SeeExample 10andExample 11. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
A solution set is an ordered triple{(x,y,z)}{(x,y,z)}that represents the intersection of three planes in space. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
A system of three equations in three variables can be solved by using a series of steps that forces a variable to be eliminated. The steps include interchanging the order of equations, multiplying both sides of an equation by a nonzero constant, and adding a nonzero multiple of one equation to another equation. SeeEx... | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
Systems of three equations in three variables are useful for solving many different types of real-world problems. SeeExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
A system of equations in three variables is inconsistent if no solution exists. After performing elimination operations, the result is a contradiction. SeeExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
Systems of equations in three variables that are inconsistent could result from three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
A system of equations in three variables is dependent if it has an infinite number of solutions. After performing elimination operations, the result is an identity. SeeExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
Systems of equations in three variables that are dependent could result from three identical planes, three planes intersecting at a line, or two identical planes that intersect the third on a line. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
There are three possible types of solutions to a system of equations representing a line and a parabola: (1) no solution, the line does not intersect the parabola; (2) one solution, the line is tangent to the parabola; and (3) two solutions, the line intersects the parabola in two points. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
There are three possible types of solutions to a system of equations representing a circle and a line: (1) no solution, the line does not intersect the circle; (2) one solution, the line is tangent to the circle; (3) two solutions, the line intersects the circle in two points. SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
There are five possible types of solutions to the system of nonlinear equations representing an ellipse and a circle:(1) no solution, the circle and the ellipse do not intersect; (2) one solution, the circle and the ellipse are tangent to each other; (3) two solutions, the circle and the ellipse intersect in two points... | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
An inequality is graphed in much the same way as an equation, except for > or <, we draw a dashed line and shade the region containing the solution set. SeeExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
Inequalities are solved the same way as equalities, but solutions to systems of inequalities must satisfy both inequalities. SeeExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
DecomposeP(x)Q(x)P(x)Q(x)by writing the partial fractions asAa1x+b1+Ba2x+b2.Aa1x+b1+Ba2x+b2.Solve by clearing the fractions, expanding the right side, collecting like terms, and setting corresponding coefficients equal to each other, then setting up and solving a system of equations. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
The decomposition ofP(x)Q(x)P(x)Q(x)with repeated linear factors must account for the factors of the denominator in increasing powers. SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
The decomposition ofP(x)Q(x)P(x)Q(x)with a nonrepeated irreducible quadratic factor needs a linear numerator over the quadratic factor, as inAx+Bx+C(ax2+bx+c).Ax+Bx+C(ax2+bx+c).SeeExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
In the decomposition ofP(x)Q(x),P(x)Q(x),whereQ(x)Q(x)has a repeated irreducible quadratic factor, when the irreducible quadratic factors are repeated, powers of the denominator factors must be represented in increasing powers asAx+B(ax2+bx+c)+A2x+B2(ax2+bx+c)2+â¯+Anx+Bn(ax2+bx+c)n.Ax+B(ax2+bx+c)+A2x+B2(ax2+bx+c)2+â¯... | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
A matrix is a rectangular array of numbers. Entries are arranged in rows and columns. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
The dimensions of a matrix refer to the number of rows and the number of columns. A3Ã23Ã2matrix has three rows and two columns. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
We add and subtract matrices of equal dimensions by adding and subtracting corresponding entries of each matrix. SeeExample 2,Example 3,Example 4, andExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
Scalar multiplication involves multiplying each entry in a matrix by a constant. SeeExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
Scalar multiplication is often required before addition or subtraction can occur. SeeExample 7. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
Multiplying matrices is possible when inner dimensions are the sameâthe number of columns in the first matrix must match the number of rows in the second. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
The product of two matrices,AAandB,B,is obtained by multiplying each entry in row 1 ofAAby each entry in column 1 ofB;B;then multiply each entry of row 1 ofAAby each entry in columns 2 ofB,B,and so on. SeeExample 8andExample 9. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
Many real-world problems can often be solved using matrices. SeeExample 10. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
We can use a calculator to perform matrix operations after saving each matrix as a matrix variable. SeeExample 11. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
An augmented matrix is one that contains the coefficients and constants of a system of equations. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
A matrix augmented with the constant column can be represented as the original system of equations. SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
Row operations include multiplying a row by a constant, adding one row to another row, and interchanging rows. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
We can use Gaussian elimination to solve a system of equations. SeeExample 3,Example 4, andExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
Row operations are performed on matrices to obtain row-echelon form. SeeExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
To solve a system of equations, write it in augmented matrix form. Perform row operations to obtain row-echelon form. Back-substitute to find the solutions. SeeExample 7andExample 8. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
A calculator can be used to solve systems of equations using matrices. SeeExample 9. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
Many real-world problems can be solved using augmented matrices. SeeExample 10andExample 11. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
An identity matrix has the propertyAI=IA=A.AI=IA=A.SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
An invertible matrix has the propertyAAâ1=Aâ1A=I.AAâ1=Aâ1A=I.SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
Use matrix multiplication and the identity to find the inverse of a2Ã22Ã2matrix. SeeExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
The multiplicative inverse can be found using a formula. SeeExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/11-key-concepts |
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