Find the angle between the two planes. $\begin{array} { l } x + y - z = 3 \\ 4 x - 5 y - 4 z = 5 \end{array}$

A) 77.7° B) 76.7° C) 79.7° D) 78.7° E) 75.7° A) 77.7° B) 76.7° C) 79.7° D) 78.7° E) 75.7°

Find u × v and show that it is orthogonal to both u and v. $\begin{array} { l } \mathbf { u } = \frac { 5 } { 7 } \mathbf { i } \\ \mathbf { v } = \frac { 6 } { 7 } \mathbf { j } - 49 \mathbf { k } \end{array}$

A) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = - 35 \mathbf { i } + \frac { 30 } { 49 } \mathbf { j } \\ ( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\ ( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \end{array}$ B) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = 35 \mathrm { i } - \frac { 30 } { 49 } \mathbf { k } \\ ( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\ ( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \end{array}$ C) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = 35 \mathbf { i } + \frac { 30 } { 49 } \mathbf { k } \\ ( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\ ( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \end{array}$ D) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = 35 \mathbf { i } + \frac { 30 } { 49 } \mathbf { j } \\ ( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\ ( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \end{array}$ E) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = 35 \mathbf { j } + \frac { 30 } { 49 } \mathbf { k } \\ ( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\ ( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \end{array}$ A) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = - 35 \mathbf { i } + \frac { 30 } { 49 } \mathbf { j } \\ ( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\ ( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \end{array}$ B) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = 35 \mathrm { i } - \frac { 30 } { 49 } \mathbf { k } \\ ( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\ ( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \end{array}$ C) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = 35 \mathbf { i } + \frac { 30 } { 49 } \mathbf { k } \\ ( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\ ( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \end{array}$ D) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = 35 \mathbf { i } + \frac { 30 } { 49 } \mathbf { j } \\ ( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\ ( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \end{array}$ E) $\begin{array} { l } \mathbf { u } \times \mathbf { v } = 35 \mathbf { j } + \frac { 30 } { 49 } \mathbf { k } \\ ( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \\ ( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \end{array}$

Exam 11: Analytic Geometry In Three Dimensions

Find the volume of the parallelepiped with the given vertices $A ( 0,0,0 ) , B ( 0,0,2 ) , C ( 0,2,0 ) , D ( 2,0,0 ) , E ( 2,2,0 ) , F ( 2,0,2 ) , G ( 0,2,2 ) , H ( 2,2,2 )$

(Multiple Choice)

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Question 101

Use vectors to determine whether the points are collinear. (-9,5,7), (-12,7,5), (3,-3,15)

(Multiple Choice)

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Question 102

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 18 x + 16 y - 10 z + 89 = 0$

(Multiple Choice)

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Question 103

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 5 x = 0$

(Multiple Choice)

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Question 104

Determine whether the planes are parallel,orthogonal,or neither. 5x - 2y - 5z = -2 X - 3y + 6z = -6

(Multiple Choice)

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Question 105

Find the angle between the two planes. x+y-z=3 4x-5y-4z=5

(Multiple Choice)

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Question 106

Find a unit vector orthogonal to u and v. $\mathbf { u } = ( 9 ) \mathbf { i } + ( - 3 ) \mathbf { j } + ( - 6 ) \mathbf { k } , \quad \mathbf { v } = ( - 2 ) \mathbf { i } + ( 8 ) \mathbf { j } + ( - 5 ) \mathbf { k }$

(Multiple Choice)

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Question 107

Use the vectors u and v to find u × (2v). $\mathbf { u } = 6 \mathbf { i } - \mathbf { j } + 7 \mathbf { k } \quad \mathbf { v } = 5 \mathbf { i } + 5 \mathbf { j } - \mathbf { k }$

(Multiple Choice)

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Question 108

Find u × v and show that it is orthogonal to both u and v. = =-49

(Multiple Choice)

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Question 109

Find the vector z,given $\mathbf { u } = ( - 1,3,4 ) , \mathbf { v } = ( 1 , - 4 , - 4 ) , \mathbf { w } = ( 20,0,20 )$ . $\mathbf { z } = 7 \mathbf { u } + \mathbf { v } - \frac { 1 } { 20 } \mathbf { w }$

(Multiple Choice)

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Question 110

Find the angle between the vectors u and v.Express your answer in degrees and round to the nearest tenth of a degree. u = $( 4 , - 6,9 )$ ,v = $( 9 , - 2,2 )$

(Multiple Choice)

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Question 111

Use the triple scalar product to find the volume of the parallelepiped having adjacent edges u,v,and w. =9+9 =9+9 =9+9 $Use the triple scalar product to find the volume of the parallelepiped having adjacent edges u,v,and w. \begin{array} { l } \mathbf { u } = 9 \mathbf { i } + 9 \mathbf { j } \\ \mathbf { v } = 9 \mathbf { j } + 9 \mathbf { k } \\ \mathbf { w } = 9 \mathbf { i } + 9 \mathbf { k } \end{array} \mathrm { a } = ( 9,9,0 ) , \mathrm { b } = ( 0,9,9 ) , \mathrm { c } = ( 9,0,9 )$ $\mathrm { a } = ( 9,9,0 ) , \mathrm { b } = ( 0,9,9 ) , \mathrm { c } = ( 9,0,9 )$

(Multiple Choice)

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Question 112

Use the vectors u and v to find u × v. $\mathbf { u } = 5 \mathbf { i } - \mathbf { j } + 6 \mathbf { k }$ $\mathbf { v } = 4 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }$

(Multiple Choice)

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Question 113

Find the general form of the equation of the plane passing through the point and perpendicular to the specified vector or line. Point: $( 2,0 , - 7 )$ Perpendicular to: $\mathbf { n } = - 7 \mathbf { k }$

(Multiple Choice)

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Question 114

Find the magnitude of the vector v described below. Initial point: (-4,-3,3) Terminal point: (8,2,-6)

(Multiple Choice)

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Question 115

Find the magnitude of the vector v described below. Initial point: (3,-7,0) Terminal point: (-1,1,0)

(Multiple Choice)

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Question 116

Find the midpoint of the line segment joining the points. $( - 5,4 , - 2 ) , ( 5,3 , - 2 )$

(Multiple Choice)

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Question 117

Determine the octant(s)in which (x,y,z)is located so that the condition(s)is (are)satisfied. $y < - 4$

(Multiple Choice)

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Question 118

Determine whether the planes are parallel,orthogonal,or neither. 6x + y - z = 2 -18x - 3y + 3z = -4

(Multiple Choice)

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Question 119

Find u × v and show that it is orthogonal to both u and v. $\mathbf { u } = ( 13,1,14 ) \quad \mathbf { v } = ( 12 , - 12 , - 1 )$

(Multiple Choice)

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Question 120

Find the volume of the parallelepiped with the given vertices $A ( 0,0,0 ) , B ( 0,0,2 ) , C ( 0,2,0 ) , D ( 2,0,0 ) , E ( 2,2,0 ) , F ( 2,0,2 ) , G ( 0,2,2 ) , H ( 2,2,2 )$

Use vectors to determine whether the points are collinear. (-9,5,7), (-12,7,5), (3,-3,15)

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 18 x + 16 y - 10 z + 89 = 0$

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 5 x = 0$

Determine whether the planes are parallel,orthogonal,or neither. 5x - 2y - 5z = -2 X - 3y + 6z = -6

Find the angle between the two planes. x+y-z=3 4x-5y-4z=5

Find a unit vector orthogonal to u and v. $\mathbf { u } = ( 9 ) \mathbf { i } + ( - 3 ) \mathbf { j } + ( - 6 ) \mathbf { k } , \quad \mathbf { v } = ( - 2 ) \mathbf { i } + ( 8 ) \mathbf { j } + ( - 5 ) \mathbf { k }$

Use the vectors u and v to find u × (2v). $\mathbf { u } = 6 \mathbf { i } - \mathbf { j } + 7 \mathbf { k } \quad \mathbf { v } = 5 \mathbf { i } + 5 \mathbf { j } - \mathbf { k }$

Find u × v and show that it is orthogonal to both u and v. = =-49

Find the vector z,given $\mathbf { u } = ( - 1,3,4 ) , \mathbf { v } = ( 1 , - 4 , - 4 ) , \mathbf { w } = ( 20,0,20 )$ . $\mathbf { z } = 7 \mathbf { u } + \mathbf { v } - \frac { 1 } { 20 } \mathbf { w }$

Find the angle between the vectors u and v.Express your answer in degrees and round to the nearest tenth of a degree. u = $( 4 , - 6,9 )$ ,v = $( 9 , - 2,2 )$

Use the vectors u and v to find u × v. $\mathbf { u } = 5 \mathbf { i } - \mathbf { j } + 6 \mathbf { k }$ $\mathbf { v } = 4 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }$

Find the general form of the equation of the plane passing through the point and perpendicular to the specified vector or line. Point: $( 2,0 , - 7 )$ Perpendicular to: $\mathbf { n } = - 7 \mathbf { k }$

Find the magnitude of the vector v described below. Initial point: (-4,-3,3) Terminal point: (8,2,-6)

Find the magnitude of the vector v described below. Initial point: (3,-7,0) Terminal point: (-1,1,0)

Find the midpoint of the line segment joining the points. $( - 5,4 , - 2 ) , ( 5,3 , - 2 )$

Determine the octant(s)in which (x,y,z)is located so that the condition(s)is (are)satisfied. $y < - 4$

Determine whether the planes are parallel,orthogonal,or neither. 6x + y - z = 2 -18x - 3y + 3z = -4

Find u × v and show that it is orthogonal to both u and v. $\mathbf { u } = ( 13,1,14 ) \quad \mathbf { v } = ( 12 , - 12 , - 1 )$

Functions and Their Graphs

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Exponential and Logarithmic Functions

Trigonometry

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Exam 11: Analytic Geometry In Three Dimensions

Use vectors to determine whether the points are collinear. (-9,5,7), (-12,7,5), (3,-3,15)

Find the center and radius of the sphere. x2+y2+z2+18x+16y−10z+89=0x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 18 x + 16 y - 10 z + 89 = 0x2+y2+z2+18x+16y−10z+89=0

Find the center and radius of the sphere. x2+y2+z2−5x=0x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 5 x = 0x2+y2+z2−5x=0

Determine whether the planes are parallel,orthogonal,or neither. 5x - 2y - 5z = -2 X - 3y + 6z = -6

Find the angle between the two planes. x+y-z=3 4x-5y-4z=5

Use the vectors u and v to find u × (2v). u=6i−j+7kv=5i+5j−k\mathbf { u } = 6 \mathbf { i } - \mathbf { j } + 7 \mathbf { k } \quad \mathbf { v } = 5 \mathbf { i } + 5 \mathbf { j } - \mathbf { k }u=6i−j+7kv=5i+5j−k

Find u × v and show that it is orthogonal to both u and v. = =-49

Find the angle between the vectors u and v.Express your answer in degrees and round to the nearest tenth of a degree. u = (4,−6,9)( 4 , - 6,9 )(4,−6,9) ,v = (9,−2,2)( 9 , - 2,2 )(9,−2,2)

Use the triple scalar product to find the volume of the parallelepiped having adjacent edges u,v,and w. =9+9 =9+9 =9+9 a=(9,9,0),b=(0,9,9),c=(9,0,9)\mathrm { a } = ( 9,9,0 ) , \mathrm { b } = ( 0,9,9 ) , \mathrm { c } = ( 9,0,9 )a=(9,9,0),b=(0,9,9),c=(9,0,9)

Use the vectors u and v to find u × v. u=5i−j+6k\mathbf { u } = 5 \mathbf { i } - \mathbf { j } + 6 \mathbf { k }u=5i−j+6k v=4i+4j−k\mathbf { v } = 4 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }v=4i+4j−k

Find the general form of the equation of the plane passing through the point and perpendicular to the specified vector or line. Point: (2,0,−7)( 2,0 , - 7 )(2,0,−7) Perpendicular to: n=−7k\mathbf { n } = - 7 \mathbf { k }n=−7k

Find the magnitude of the vector v described below. Initial point: (-4,-3,3) Terminal point: (8,2,-6)

Find the magnitude of the vector v described below. Initial point: (3,-7,0) Terminal point: (-1,1,0)

Find the midpoint of the line segment joining the points. (−5,4,−2),(5,3,−2)( - 5,4 , - 2 ) , ( 5,3 , - 2 )(−5,4,−2),(5,3,−2)

Determine the octant(s)in which (x,y,z)is located so that the condition(s)is (are)satisfied. y<−4y < - 4y<−4

Determine whether the planes are parallel,orthogonal,or neither. 6x + y - z = 2 -18x - 3y + 3z = -4

Find u × v and show that it is orthogonal to both u and v. u=(13,1,14)v=(12,−12,−1)\mathbf { u } = ( 13,1,14 ) \quad \mathbf { v } = ( 12 , - 12 , - 1 )u=(13,1,14)v=(12,−12,−1)

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Analytic Trigonometry

Additional Topics In Trigonometery

Systems Of Equations and Inequalities

Matrices and Determinants

Sequences Series and Probability

Topics In Analytic Geometry

Limits and An Introduction To Calculus

Filters

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 18 x + 16 y - 10 z + 89 = 0$

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 5 x = 0$

Use the vectors u and v to find u × (2v). $\mathbf { u } = 6 \mathbf { i } - \mathbf { j } + 7 \mathbf { k } \quad \mathbf { v } = 5 \mathbf { i } + 5 \mathbf { j } - \mathbf { k }$

Find the angle between the vectors u and v.Express your answer in degrees and round to the nearest tenth of a degree. u = $( 4 , - 6,9 )$ ,v = $( 9 , - 2,2 )$

Use the vectors u and v to find u × v. $\mathbf { u } = 5 \mathbf { i } - \mathbf { j } + 6 \mathbf { k }$ $\mathbf { v } = 4 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }$

Find the general form of the equation of the plane passing through the point and perpendicular to the specified vector or line. Point: $( 2,0 , - 7 )$ Perpendicular to: $\mathbf { n } = - 7 \mathbf { k }$

Find the midpoint of the line segment joining the points. $( - 5,4 , - 2 ) , ( 5,3 , - 2 )$

Determine the octant(s)in which (x,y,z)is located so that the condition(s)is (are)satisfied. $y < - 4$

Find u × v and show that it is orthogonal to both u and v. $\mathbf { u } = ( 13,1,14 ) \quad \mathbf { v } = ( 12 , - 12 , - 1 )$