Question 1

Find the magnitude of the vector $\mathbf { v }$.
$$\mathbf { v } = \langle - 7,1 , - 8 \rangle$$

Accepted Answer

A)  $- 14$ 
B)  16 
C)  $\sqrt { 14 }$ 
D)  $2 \sqrt { 114 }$ 
E)  $\sqrt { 114 }$ 
A)  $- 14$ 
B)  16 
C)  $\sqrt { 14 }$ 
D)  $2 \sqrt { 114 }$ 
E)  $\sqrt { 114 }$

Question 2

Find the angle between the two planes in degrees. Round to a tenth of a degree.
$$\begin{array} { l } 
2 x - y - 4 z = 4 \
- 3 x - 2 y + 4 z = 4
\end{array}$$

Accepted Answer

A)  $11.5 ^ { \circ }$ 
B)  $54.1 ^ { \circ }$ 
C)  $144.1 ^ { \circ }$ 
D)  $39.0 ^ { \circ }$ 
E)  $2.5 ^ { \circ }$ 
A)  $11.5 ^ { \circ }$ 
B)  $54.1 ^ { \circ }$ 
C)  $144.1 ^ { \circ }$ 
D)  $39.0 ^ { \circ }$ 
E)  $2.5 ^ { \circ }$

Question 3

Find the vector $\mathbf { z }$, given $\mathbf { u } = \langle 3 , - 3,8 \rangle , \mathbf { v } = \langle - 7 , - 5,2 \rangle$, and $\mathbf { w } = \langle 8 , - 24,18 \rangle$. $- \mathbf { u } + 3 \mathbf { v } - 4 \mathbf { z } = \mathbf { w }$

Accepted Answer

A)  $\langle 32 , - 12,20 \rangle$ 
B)  $\langle 3 , - 8 , - 5 \rangle$ 
C)  $\langle 8 , - 3 , - 5 \rangle$ 
D)  $\langle - 8,3 , - 5 \rangle$ 
E)  $\langle - 7,3 , - 4 \rangle$ 
A)  $\langle 32 , - 12,20 \rangle$ 
B)  $\langle 3 , - 8 , - 5 \rangle$ 
C)  $\langle 8 , - 3 , - 5 \rangle$ 
D)  $\langle - 8,3 , - 5 \rangle$ 
E)  $\langle - 7,3 , - 4 \rangle$

Question 4

Find the general form of the equation of the plane with the given characteristics.
The plane passes through the points $( 4,5 , - 2 )$ and $( - 1 , - 4 , - 1 )$ and is perpendicular to the plane $2 x + 5 y + 3 z = - 3$.

Accepted Answer

A)  $32 x - 17 y + 7 z - 29 = 0$ 
B)  $2 x + 5 y + 3 z + 27 = 0$ 
C)  $32 x + 17 y + 7 z + 199 = 0$ 
D)  $2 x + 5 y + 3 z = 0$ 
E)  $33 x - 16 y + 5 z - 42 = 0$ 
A)  $32 x - 17 y + 7 z - 29 = 0$ 
B)  $2 x + 5 y + 3 z + 27 = 0$ 
C)  $32 x + 17 y + 7 z + 199 = 0$ 
D)  $2 x + 5 y + 3 z = 0$ 
E)  $33 x - 16 y + 5 z - 42 = 0$

Question 5

Find the center and radius of the sphere.
$$x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 16 x + 6 y + 6 z + 18 = 0$$

Accepted Answer

A)  center: $( - 8,3,3 )$; radius: 8 
B)  center: $( 8 , - 3 , - 3 )$; radius: 8 
C)  center: $( 8,3,3 )$; radius: 8 
D)  center: $( - 8,3,3 )$; radius: 64 
E)  center: $( 8 , - 3 , - 3 )$; radius: 64 
A)  center: $( - 8,3,3 )$; radius: 8 
B)  center: $( 8 , - 3 , - 3 )$; radius: 8 
C)  center: $( 8,3,3 )$; radius: 8 
D)  center: $( - 8,3,3 )$; radius: 64 
E)  center: $( 8 , - 3 , - 3 )$; radius: 64

Question 6

Find a set of parametric equations for the line through the point and parallel to the specified line. Show all your work.
$$x = 2 - 5 t$$
$( 3 , - 3,9 )$, parallel to $y = - 9 - 3 t$
$$z = 8 - 2 t$$

Accepted Answer

Answers may vary. On

Question 7

Use the scalar triple product to find the volume of the parallelepiped having adjacent edges $\langle 1,2,2 \rangle , \langle 2,3,1 \rangle$, and $\langle 1,3,4 \rangle$.

Accepted Answer

A)  29 
B)  15 
C)  1 
D)  51 
E)  28 
A)  29 
B)  15 
C)  1 
D)  51 
E)  28

Question 8

Find the angle between the two planes in degrees. Round to a tenth of a degree.
$$\begin{array} { l } 
x + 3 y + 6 z = 4 \
x - 6 y - z = 4
\end{array}$$

Accepted Answer

A)  $9.4 ^ { \circ }$ 
B)  $33.4 ^ { \circ }$ 
C)  $123.4 ^ { \circ }$ 
D)  $28.8 ^ { \circ }$ 
E)  $2.2 ^ { \circ }$ 
A)  $9.4 ^ { \circ }$ 
B)  $33.4 ^ { \circ }$ 
C)  $123.4 ^ { \circ }$ 
D)  $28.8 ^ { \circ }$ 
E)  $2.2 ^ { \circ }$

Question 9

Find the magnitude of the vector described below.
Initial point: $( 0 , - 6 , - 4 )$
Terminal point: $( - 1,5,5 )$

Accepted Answer

A)  19 
B)  50 
C)  $\sqrt { 21 }$ 
D)  $2 \sqrt { 203 }$ 
E)  $\sqrt { 203 }$ 
A)  19 
B)  50 
C)  $\sqrt { 21 }$ 
D)  $2 \sqrt { 203 }$ 
E)  $\sqrt { 203 }$

Question 10

Find the magnitude of the vector $\mathbf { v }$.
$$\mathbf { v } = \langle - 4,0 , - 8 \rangle$$

Accepted Answer

A)  $- 12$ 
B)  12 
C)  $2 \sqrt { 3 }$ 
D)  $8 \sqrt { 5 }$ 
E)  $4 \sqrt { 5 }$ 
A)  $- 12$ 
B)  12 
C)  $2 \sqrt { 3 }$ 
D)  $8 \sqrt { 5 }$ 
E)  $4 \sqrt { 5 }$

Question 11

Find the angle between the vectors $\mathbf { u }$ and $\mathbf { v }$. Express your answer in degrees and round to the nearest tenth of a degree.
$$\mathbf { u } = \langle - 3,6,3 \rangle , \mathbf { v } = \langle 9,9,9 \rangle$$

Accepted Answer

A)  $25.2 ^ { \circ }$ 
B)  $28.1 ^ { \circ }$ 
C)  $64.8 ^ { \circ }$ 
D)  $90 ^ { \circ }$ 
E)  $61.9 ^ { \circ }$ 
A)  $25.2 ^ { \circ }$ 
B)  $28.1 ^ { \circ }$ 
C)  $64.8 ^ { \circ }$ 
D)  $90 ^ { \circ }$ 
E)  $61.9 ^ { \circ }$

Question 12

Write the component form of the vector described below.
Initial point: $( - 7,2 , - 1 )$
Terminal point: $( - 5 , - 9 , - 6 )$

Accepted Answer

A)  $\langle 2 , - 11 , - 5 \rangle$ 
B)  $\langle - 2,11,5 \rangle$ 
C)  $\langle - 5 , - 9 , - 6 \rangle$ 
D)  $\langle 2 , - 11 , - 7 \rangle$ 
E)  $\langle 35 , - 18,6 \rangle$ 
A)  $\langle 2 , - 11 , - 5 \rangle$ 
B)  $\langle - 2,11,5 \rangle$ 
C)  $\langle - 5 , - 9 , - 6 \rangle$ 
D)  $\langle 2 , - 11 , - 7 \rangle$ 
E)  $\langle 35 , - 18,6 \rangle$

Question 13

Find the distance between the points.
$$( 2 , - 3,3 ) , ( - 6,2,7 )$$

Accepted Answer

A)  5 
B)  17 
C)  $\sqrt { 17 }$ 
D)  $2 \sqrt { 105 }$ 
E)  $\sqrt { 105 }$ 
A)  5 
B)  17 
C)  $\sqrt { 17 }$ 
D)  $2 \sqrt { 105 }$ 
E)  $\sqrt { 105 }$

Question 14

Determine whether the planes are parallel, orthogonal, or neither.
$$\begin{array} { l } 
2 x - 3 y - 6 z = - 4 \
8 x - 12 y - 24 z = - 14
\end{array}$$

Accepted Answer

A)  parallel 
B)  orthogonal 
C)  neither 
A)  parallel 
B)  orthogonal 
C)  neither

Question 15

Find a set of parametric equations for the line that passes through the given points.
Show all your work.
$$( - 4,2,8 ) , ( 5 , - 2 , - 7 )$$

Accepted Answer

Answers may vary. On

Question 16

Determine whether the planes are parallel, orthogonal, or neither.
$$\begin{array} { l } 
6 x - y - z = 4 \
24 x - 4 y - 4 z = 18
\end{array}$$

Accepted Answer

A)  parallel 
B)  orthogonal 
C)  neither 
A)  parallel 
B)  orthogonal 
C)  neither

Question 17

Find $\mathbf { u } \times \mathbf { v }$.
$$\mathbf { u } = \langle 3 , - 7 , - 1 \rangle , \mathbf { v } = \langle - 9,4 , - 4 \rangle$$

Accepted Answer

A)  $\langle 32,21 , - 51 \rangle$ 
B)  $- 51$ 
C)  5 
D)  $\langle 32 , - 21 , - 51 \rangle$ 
E)  $\langle - 27 , - 28,4 \rangle$ 
A)  $\langle 32,21 , - 51 \rangle$ 
B)  $- 51$ 
C)  5 
D)  $\langle 32 , - 21 , - 51 \rangle$ 
E)  $\langle - 27 , - 28,4 \rangle$

Question 18

Determine the values of $c$ such that $\| c \mathbf { u } \| = 6$, where $\mathbf { u } = 6 \mathbf { i } + 5 \mathbf { j } + 5 \mathbf { k }$.

Accepted Answer

A)  $c = \pm 6$ 
B)  $c = \pm \frac { 3 \sqrt { 86 } } { 43 }$ 
C)  $c = \pm \frac { \sqrt { 86 } } { 6 }$ 
D)  $c = \pm \frac { 1 } { 6 }$ 
E)  $c = \pm \frac { \sqrt { 86 } } { 86 }$ 
A)  $c = \pm 6$ 
B)  $c = \pm \frac { 3 \sqrt { 86 } } { 43 }$ 
C)  $c = \pm \frac { \sqrt { 86 } } { 6 }$ 
D)  $c = \pm \frac { 1 } { 6 }$ 
E)  $c = \pm \frac { \sqrt { 86 } } { 86 }$

Question 19

Find symmetric equations for the line through the point and parallel to the specified line. Show all your work.
$$x = - 8 + 7 t$$
$( - 5 , - 3,3 )$, parallel to
$$\begin{array} { c } 
y = 7 + 3 t \
z = - 9 - 5 t
\end{array}$$

Accepted Answer

Answers may vary. On

Question 20

Find the magnitude of the vector $\mathbf { v }$.
$$\mathbf { v } = \langle - 3 , - 3 , - 5 \rangle$$

Accepted Answer

A)  $- 11$ 
B)  11 
C)  $\sqrt { 11 }$ 
D)  $2 \sqrt { 43 }$ 
E)  $\sqrt { 43 }$ 
A)  $- 11$ 
B)  11 
C)  $\sqrt { 11 }$ 
D)  $2 \sqrt { 43 }$ 
E)  $\sqrt { 43 }$

Find the magnitude of the vector $\mathbf { v }$ . $\mathbf { v } = \langle - 7,1 , - 8 \rangle$

Find the angle between the two planes in degrees. Round to a tenth of a degree. 2x-y-4z=4 -3x-2y+4z=4

Find the vector $\mathbf { z }$ , given $\mathbf { u } = \langle 3 , - 3,8 \rangle , \mathbf { v } = \langle - 7 , - 5,2 \rangle$ , and $\mathbf { w } = \langle 8 , - 24,18 \rangle$ . $- \mathbf { u } + 3 \mathbf { v } - 4 \mathbf { z } = \mathbf { w }$

Find the general form of the equation of the plane with the given characteristics. The plane passes through the points $( 4,5 , - 2 )$ and $( - 1 , - 4 , - 1 )$ and is perpendicular to the plane $2 x + 5 y + 3 z = - 3$ .

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 16 x + 6 y + 6 z + 18 = 0$

Find a set of parametric equations for the line through the point and parallel to the specified line. Show all your work. $x = 2 - 5 t$ $( 3 , - 3,9 )$ , parallel to $y = - 9 - 3 t$ $z = 8 - 2 t$

Use the scalar triple product to find the volume of the parallelepiped having adjacent edges $\langle 1,2,2 \rangle , \langle 2,3,1 \rangle$ , and $\langle 1,3,4 \rangle$ .

Find the angle between the two planes in degrees. Round to a tenth of a degree. x+3y+6z=4 x-6y-z=4

Find the magnitude of the vector described below. Initial point: $( 0 , - 6 , - 4 )$ Terminal point: $( - 1,5,5 )$

Find the magnitude of the vector $\mathbf { v }$ . $\mathbf { v } = \langle - 4,0 , - 8 \rangle$

Find the angle between the vectors $\mathbf { u }$ and $\mathbf { v }$ . Express your answer in degrees and round to the nearest tenth of a degree. $\mathbf { u } = \langle - 3,6,3 \rangle , \mathbf { v } = \langle 9,9,9 \rangle$

Write the component form of the vector described below. Initial point: $( - 7,2 , - 1 )$ Terminal point: $( - 5 , - 9 , - 6 )$

Find the distance between the points. $( 2 , - 3,3 ) , ( - 6,2,7 )$

Determine whether the planes are parallel, orthogonal, or neither. 2x-3y-6z=-4 8x-12y-24z=-14

Find a set of parametric equations for the line that passes through the given points. Show all your work. $( - 4,2,8 ) , ( 5 , - 2 , - 7 )$

Determine whether the planes are parallel, orthogonal, or neither. 6x-y-z=4 24x-4y-4z=18

Find $\mathbf { u } \times \mathbf { v }$ . $\mathbf { u } = \langle 3 , - 7 , - 1 \rangle , \mathbf { v } = \langle - 9,4 , - 4 \rangle$

Determine the values of $c$ such that $\| c \mathbf { u } \| = 6$ , where $\mathbf { u } = 6 \mathbf { i } + 5 \mathbf { j } + 5 \mathbf { k }$ .

Find symmetric equations for the line through the point and parallel to the specified line. Show all your work. $x = - 8 + 7 t$ $( - 5 , - 3,3 )$ , parallel to y=7+3t z=-9-5t

Find the magnitude of the vector $\mathbf { v }$ . $\mathbf { v } = \langle - 3 , - 3 , - 5 \rangle$

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Limits and an Introduction to Calculus

Filters

Exam 10: Analytic Geometry in Three Dimensions

Find the magnitude of the vector v\mathbf { v }v . v=⟨−7,1,−8⟩\mathbf { v } = \langle - 7,1 , - 8 \ranglev=⟨−7,1,−8⟩

Find the angle between the two planes in degrees. Round to a tenth of a degree. 2x-y-4z=4 -3x-2y+4z=4

Find the general form of the equation of the plane with the given characteristics. The plane passes through the points (4,5,−2)( 4,5 , - 2 )(4,5,−2) and (−1,−4,−1)( - 1 , - 4 , - 1 )(−1,−4,−1) and is perpendicular to the plane 2x+5y+3z=−32 x + 5 y + 3 z = - 32x+5y+3z=−3 .

Find the center and radius of the sphere. x2+y2+z2−16x+6y+6z+18=0x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 16 x + 6 y + 6 z + 18 = 0x2+y2+z2−16x+6y+6z+18=0

Find a set of parametric equations for the line through the point and parallel to the specified line. Show all your work. x=2−5tx = 2 - 5 tx=2−5t (3,−3,9)( 3 , - 3,9 )(3,−3,9) , parallel to y=−9−3ty = - 9 - 3 ty=−9−3t z=8−2tz = 8 - 2 tz=8−2t

Use the scalar triple product to find the volume of the parallelepiped having adjacent edges ⟨1,2,2⟩,⟨2,3,1⟩\langle 1,2,2 \rangle , \langle 2,3,1 \rangle⟨1,2,2⟩,⟨2,3,1⟩ , and ⟨1,3,4⟩\langle 1,3,4 \rangle⟨1,3,4⟩ .

Find the angle between the two planes in degrees. Round to a tenth of a degree. x+3y+6z=4 x-6y-z=4

Find the magnitude of the vector described below. Initial point: (0,−6,−4)( 0 , - 6 , - 4 )(0,−6,−4) Terminal point: (−1,5,5)( - 1,5,5 )(−1,5,5)

Find the magnitude of the vector v\mathbf { v }v . v=⟨−4,0,−8⟩\mathbf { v } = \langle - 4,0 , - 8 \ranglev=⟨−4,0,−8⟩

Find the angle between the vectors u\mathbf { u }u and v\mathbf { v }v . Express your answer in degrees and round to the nearest tenth of a degree. u=⟨−3,6,3⟩,v=⟨9,9,9⟩\mathbf { u } = \langle - 3,6,3 \rangle , \mathbf { v } = \langle 9,9,9 \rangleu=⟨−3,6,3⟩,v=⟨9,9,9⟩

Write the component form of the vector described below. Initial point: (−7,2,−1)( - 7,2 , - 1 )(−7,2,−1) Terminal point: (−5,−9,−6)( - 5 , - 9 , - 6 )(−5,−9,−6)

Find the distance between the points. (2,−3,3),(−6,2,7)( 2 , - 3,3 ) , ( - 6,2,7 )(2,−3,3),(−6,2,7)

Determine whether the planes are parallel, orthogonal, or neither. 2x-3y-6z=-4 8x-12y-24z=-14

Find a set of parametric equations for the line that passes through the given points. Show all your work. (−4,2,8),(5,−2,−7)( - 4,2,8 ) , ( 5 , - 2 , - 7 )(−4,2,8),(5,−2,−7)

Determine whether the planes are parallel, orthogonal, or neither. 6x-y-z=4 24x-4y-4z=18

Find u×v\mathbf { u } \times \mathbf { v }u×v . u=⟨3,−7,−1⟩,v=⟨−9,4,−4⟩\mathbf { u } = \langle 3 , - 7 , - 1 \rangle , \mathbf { v } = \langle - 9,4 , - 4 \rangleu=⟨3,−7,−1⟩,v=⟨−9,4,−4⟩

Determine the values of ccc such that ∥cu∥=6\| c \mathbf { u } \| = 6∥cu∥=6 , where u=6i+5j+5k\mathbf { u } = 6 \mathbf { i } + 5 \mathbf { j } + 5 \mathbf { k }u=6i+5j+5k .

Find symmetric equations for the line through the point and parallel to the specified line. Show all your work. x=−8+7tx = - 8 + 7 tx=−8+7t (−5,−3,3)( - 5 , - 3,3 )(−5,−3,3) , parallel to y=7+3t z=-9-5t

Find the magnitude of the vector v\mathbf { v }v . v=⟨−3,−3,−5⟩\mathbf { v } = \langle - 3 , - 3 , - 5 \ranglev=⟨−3,−3,−5⟩

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Topics in Analytic Geometry

Limits and an Introduction to Calculus

Filters

Find the magnitude of the vector $\mathbf { v }$ . $\mathbf { v } = \langle - 7,1 , - 8 \rangle$

Find the general form of the equation of the plane with the given characteristics. The plane passes through the points $( 4,5 , - 2 )$ and $( - 1 , - 4 , - 1 )$ and is perpendicular to the plane $2 x + 5 y + 3 z = - 3$ .

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 16 x + 6 y + 6 z + 18 = 0$

Find a set of parametric equations for the line through the point and parallel to the specified line. Show all your work. $x = 2 - 5 t$ $( 3 , - 3,9 )$ , parallel to $y = - 9 - 3 t$ $z = 8 - 2 t$

Use the scalar triple product to find the volume of the parallelepiped having adjacent edges $\langle 1,2,2 \rangle , \langle 2,3,1 \rangle$ , and $\langle 1,3,4 \rangle$ .

Find the magnitude of the vector described below. Initial point: $( 0 , - 6 , - 4 )$ Terminal point: $( - 1,5,5 )$

Find the magnitude of the vector $\mathbf { v }$ . $\mathbf { v } = \langle - 4,0 , - 8 \rangle$

Find the angle between the vectors $\mathbf { u }$ and $\mathbf { v }$ . Express your answer in degrees and round to the nearest tenth of a degree. $\mathbf { u } = \langle - 3,6,3 \rangle , \mathbf { v } = \langle 9,9,9 \rangle$

Write the component form of the vector described below. Initial point: $( - 7,2 , - 1 )$ Terminal point: $( - 5 , - 9 , - 6 )$

Find the distance between the points. $( 2 , - 3,3 ) , ( - 6,2,7 )$

Find a set of parametric equations for the line that passes through the given points. Show all your work. $( - 4,2,8 ) , ( 5 , - 2 , - 7 )$

Find $\mathbf { u } \times \mathbf { v }$ . $\mathbf { u } = \langle 3 , - 7 , - 1 \rangle , \mathbf { v } = \langle - 9,4 , - 4 \rangle$

Determine the values of $c$ such that $\| c \mathbf { u } \| = 6$ , where $\mathbf { u } = 6 \mathbf { i } + 5 \mathbf { j } + 5 \mathbf { k }$ .

Find symmetric equations for the line through the point and parallel to the specified line. Show all your work. $x = - 8 + 7 t$ $( - 5 , - 3,3 )$ , parallel to y=7+3t z=-9-5t

Find the magnitude of the vector $\mathbf { v }$ . $\mathbf { v } = \langle - 3 , - 3 , - 5 \rangle$