Question 1

Find the midpoint of the line segment joining the points.
$$( - 5,8,9 ) , ( - 8,9,2 )$$

Accepted Answer

A)  $\left( \frac { 3 } { 2 } , \frac { - 1 } { 2 } , \frac { 7 } { 2 } \right)$ 
B)  $\left( \frac { - 13 } { 2 } , \frac { 17 } { 2 } , \frac { 11 } { 2 } \right)$ 
C)  $( - 13,17,11 )$ 
D)  $( 3 , - 1,7 )$ 
E)  $( 40,72,18 )$ 
A)  $\left( \frac { 3 } { 2 } , \frac { - 1 } { 2 } , \frac { 7 } { 2 } \right)$ 
B)  $\left( \frac { - 13 } { 2 } , \frac { 17 } { 2 } , \frac { 11 } { 2 } \right)$ 
C)  $( - 13,17,11 )$ 
D)  $( 3 , - 1,7 )$ 
E)  $( 40,72,18 )$ B

Question 2

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 A

Question 3

Find the vector $\mathbf { z }$, given $\mathbf { u } = \langle - 3,3,9 \rangle$ and $\mathbf { v } = \langle - 7,5,7 \rangle$. $\mathbf { z } = - 3 \mathbf { u } - 5 \mathbf { v }$

Accepted Answer

A)  $\langle 2 , - 4 , - 20 \rangle$ 
B)  $\langle 44 , - 34 , - 62 \rangle$ 
C)  $\langle - 10,8,16 \rangle$ 
D)  $\langle 36 , - 30 , - 66 \rangle$ 
E)  $\langle - 26,16,8 \rangle$ 
A)  $\langle 2 , - 4 , - 20 \rangle$ 
B)  $\langle 44 , - 34 , - 62 \rangle$ 
C)  $\langle - 10,8,16 \rangle$ 
D)  $\langle 36 , - 30 , - 66 \rangle$ 
E)  $\langle - 26,16,8 \rangle$ B

Question 4

Find the vector $\mathbf { z }$, given $\mathbf { u } = \langle 0 , - 8 , - 4 \rangle , \mathbf { v } = \langle - 5,0 , - 6 \rangle$, and $\mathbf { w } = \langle - 17,25 , - 1 \rangle$. $- 2 \mathbf { u } + 4 \mathbf { v } - 3 \mathbf { z } = \mathbf { w }$

Accepted Answer

A)  $\langle 3,9,15 \rangle$ 
B)  $\langle - 3 , - 1 , - 5 \rangle$ 
C)  $\langle 1,3 , - 5 \rangle$ 
D)  $\langle - 1 , - 3 , - 5 \rangle$ 
E)  $\langle 0 , - 3 , - 4 \rangle$ 
A)  $\langle 3,9,15 \rangle$ 
B)  $\langle - 3 , - 1 , - 5 \rangle$ 
C)  $\langle 1,3 , - 5 \rangle$ 
D)  $\langle - 1 , - 3 , - 5 \rangle$ 
E)  $\langle 0 , - 3 , - 4 \rangle$

Question 5

Find the magnitude of the vector $\mathbf { v }$.
$$\mathbf { v } = \langle 0 , - 5 , - 3 \rangle$$

Accepted Answer

A)  $- 8$ 
B)  8 
C)  $2 \sqrt { 2 }$ 
D)  $2 \sqrt { 34 }$ 
E)  $\sqrt { 34 }$ 
A)  $- 8$ 
B)  8 
C)  $2 \sqrt { 2 }$ 
D)  $2 \sqrt { 34 }$ 
E)  $\sqrt { 34 }$

Question 6

Find a set of parametric equations for the line that passes through the given points. Show all your work.
$$( 4 , - 7,2 ) , ( - 7 , - 1 , - 5 )$$

Accepted Answer

Answers may vary. On

Question 7

Write the component form of the vector described below.
Initial point: $( - 1 , - 5 , - 6 )$
Terminal point: $( 4 , - 2 , - 4 )$

Accepted Answer

A)  $\langle 5,3,2 \rangle$ 
B)  $\langle - 5 , - 3 , - 2 \rangle$ 
C)  $\langle 4 , - 2 , - 4 \rangle$ 
D)  $\langle 5,3 , - 10 \rangle$ 
E)  $\langle - 4,10,24 \rangle$ 
A)  $\langle 5,3,2 \rangle$ 
B)  $\langle - 5 , - 3 , - 2 \rangle$ 
C)  $\langle 4 , - 2 , - 4 \rangle$ 
D)  $\langle 5,3 , - 10 \rangle$ 
E)  $\langle - 4,10,24 \rangle$

Question 8

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 9

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 10

Find the triple scalar product $\mathbf { u } \cdot ( \mathbf { v } \times \mathbf { w } )$ for the vectors $\mathbf { u } = - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } , \mathbf { v } = 7 \mathbf { i } + 8 \mathbf { j } + 3 \mathbf { k } , \mathbf { w } = 4 \mathbf { i } + 7 \mathbf { j } - 3 \mathbf { k }$

Accepted Answer

A)  75 
B)  128 
C)  $- 128$ 
D)  $- 4$ 
E)  0 
A)  75 
B)  128 
C)  $- 128$ 
D)  $- 4$ 
E)  0

Question 11

Find the area of the parallelogram formed by the points $A ( 2 , - 3,5 ) , B ( 7 , - 2,7 )$, $C ( 3 , - 2,11 )$, and $D ( 8 , - 1,13 )$.

Accepted Answer

A)  816 
B)  $2 \sqrt { 5 }$ 
C)  20 
D)  $4 \sqrt { 51 }$ 
E)  18 
A)  816 
B)  $2 \sqrt { 5 }$ 
C)  20 
D)  $4 \sqrt { 51 }$ 
E)  18

Question 12

Find a unit vector orthogonal to $\mathbf { u }$ and $\mathbf { v }$.
$\mathbf { u } =$ leadcoeff(a)i $\operatorname { coeff } ( b ) \mathbf { j } \operatorname { coeff } ( \mathrm { c } ) \mathbf { k } , \mathbf { v } =$ leadcoeff $( \mathrm { d } ) \mathbf { i } \operatorname { coeff } ( \mathrm { f } ) \mathbf { j } \operatorname { coeff } ( \mathrm { g } ) \mathbf { k }$

Accepted Answer

A)  $\frac { 1 } { \text { leadcoeff(da 1 fac } ) \sqrt { \text { da 1 rad } } } ($ leadcoeff(a* $\left. d ) \mathbf { i } \operatorname { coeff } \left( b ^ { * } f \right) \mathbf { j } \operatorname { coeff } \left( c ^ { * } g \right) \mathbf { k } \right)$ 
B)  $\frac { 1 } { \text { leadcoeff(da } 2 \text { fac } ) \sqrt { \text { da } 2 \text { rad } } } \left( \right.$ leadcoeff $\left. \left( a ^ { \wedge } 2 \right) \mathbf { i } \operatorname { coeff } \left( b ^ { \wedge } 2 \right) \mathbf { j } \operatorname { coeff } \left( c ^ { \wedge } 2 \right) \mathbf { k } \right)$ 
C)  leadcoeff(b* $\left. g - c ^ { * } f \right) \mathbf { i } \operatorname { coeff } \left( - a ^ { * } g + c ^ { * } d \right) \mathbf { j } \operatorname { coeff } \left( a ^ { * } f - d ^ { * } b \right) \mathbf { k }$ 
D)  $\frac { 1 } { \text { leadcoeff(wfac) } \sqrt { w r a d } } \left( \right.$ leadcoeff $\left. \left( b ^ { * } g - c ^ { * } f \right) \mathbf { i } \operatorname { coeff } \left( - a ^ { * } g + c ^ { * } d \right) \mathbf { j } \operatorname { coeff } \left( a ^ { * } f - d ^ { * } b \right) \mathbf { k } \right)$ 
E)  leadcoeff( $\left. a ^ { * } d \right) \mathbf { i } \operatorname { coeff } \left( b ^ { * } f \right) \mathbf { j } \operatorname { coeff } \left( c ^ { * } g \right) \mathbf { k }$ 
A)  $\frac { 1 } { \text { leadcoeff(da 1 fac } ) \sqrt { \text { da 1 rad } } } ($ leadcoeff(a* $\left. d ) \mathbf { i } \operatorname { coeff } \left( b ^ { * } f \right) \mathbf { j } \operatorname { coeff } \left( c ^ { * } g \right) \mathbf { k } \right)$ 
B)  $\frac { 1 } { \text { leadcoeff(da } 2 \text { fac } ) \sqrt { \text { da } 2 \text { rad } } } \left( \right.$ leadcoeff $\left. \left( a ^ { \wedge } 2 \right) \mathbf { i } \operatorname { coeff } \left( b ^ { \wedge } 2 \right) \mathbf { j } \operatorname { coeff } \left( c ^ { \wedge } 2 \right) \mathbf { k } \right)$ 
C)  leadcoeff(b* $\left. g - c ^ { * } f \right) \mathbf { i } \operatorname { coeff } \left( - a ^ { * } g + c ^ { * } d \right) \mathbf { j } \operatorname { coeff } \left( a ^ { * } f - d ^ { * } b \right) \mathbf { k }$ 
D)  $\frac { 1 } { \text { leadcoeff(wfac) } \sqrt { w r a d } } \left( \right.$ leadcoeff $\left. \left( b ^ { * } g - c ^ { * } f \right) \mathbf { i } \operatorname { coeff } \left( - a ^ { * } g + c ^ { * } d \right) \mathbf { j } \operatorname { coeff } \left( a ^ { * } f - d ^ { * } b \right) \mathbf { k } \right)$ 
E)  leadcoeff( $\left. a ^ { * } d \right) \mathbf { i } \operatorname { coeff } \left( b ^ { * } f \right) \mathbf { j } \operatorname { coeff } \left( c ^ { * } g \right) \mathbf { k }$

Question 13

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 14

Find a set of parametric equations for the line through the point and parallel to the specified vector. Show all your work.
$( - 1 , - 5,9 )$, parallel to $\langle - 7,6 , - 3 \rangle$

Accepted Answer

Answers may vary. On

Question 15

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 16

Find the general form of the equation of the plane passing through the three points. [Be sure to reduce the coefficients in your answer to lowest terms by dividing out any common factor.]
$$( 5 , - 5 , - 6 ) , ( 6 , - 1,4 ) , ( - 2 , - 5,5 )$$

Accepted Answer

A)  $44 x - 81 y + 28 z - 457 = 0$ 
B)  $44 x - 81 y + 28 z = 0$ 
C)  $5 x - 5 y - 6 z = 0$ 
D)  $44 x - 81 y + 28 z + 457 = 0$ 
E)  $5 x - 5 y - 6 z + 457 = 0$ 
A)  $44 x - 81 y + 28 z - 457 = 0$ 
B)  $44 x - 81 y + 28 z = 0$ 
C)  $5 x - 5 y - 6 z = 0$ 
D)  $44 x - 81 y + 28 z + 457 = 0$ 
E)  $5 x - 5 y - 6 z + 457 = 0$

Question 17

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 18

Find a set of parametric equations for the line that passes through the given points.
Show all your work.
$$\left( 4 , \frac { 9 } { 2 } , \frac { - 3 } { 2 } \right) , \left( \frac { 1 } { 2 } , \frac { - 3 } { 2 } , - 2 \right)$$

Accepted Answer

Answers may vary. On

Question 19

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.
$$\mathbf { u } = - 6 \mathbf { i } + 9 \mathbf { j } + 2 \mathbf { k } , \mathbf { v } = - 3 \mathbf { i } - 7 \mathbf { j } - 4 \mathbf { k }$$

Accepted Answer

A)  $29.3 ^ { \circ }$ 
B)  $34.1 ^ { \circ }$ 
C)  $60.7 ^ { \circ }$ 
D)  $90 ^ { \circ }$ 
E)  $124.1 ^ { \circ }$ 
A)  $29.3 ^ { \circ }$ 
B)  $34.1 ^ { \circ }$ 
C)  $60.7 ^ { \circ }$ 
D)  $90 ^ { \circ }$ 
E)  $124.1 ^ { \circ }$

Question 20

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

Find the midpoint of the line segment joining the points. $( - 5,8,9 ) , ( - 8,9,2 )$

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

Find the vector $\mathbf { z }$ , given $\mathbf { u } = \langle - 3,3,9 \rangle$ and $\mathbf { v } = \langle - 7,5,7 \rangle$ . $\mathbf { z } = - 3 \mathbf { u } - 5 \mathbf { v }$

Find the vector $\mathbf { z }$ , given $\mathbf { u } = \langle 0 , - 8 , - 4 \rangle , \mathbf { v } = \langle - 5,0 , - 6 \rangle$ , and $\mathbf { w } = \langle - 17,25 , - 1 \rangle$ . $- 2 \mathbf { u } + 4 \mathbf { v } - 3 \mathbf { z } = \mathbf { w }$

Find the magnitude of the vector $\mathbf { v }$ . $\mathbf { v } = \langle 0 , - 5 , - 3 \rangle$

Find a set of parametric equations for the line that passes through the given points. Show all your work. $( 4 , - 7,2 ) , ( - 7 , - 1 , - 5 )$

Write the component form of the vector described below. Initial point: $( - 1 , - 5 , - 6 )$ Terminal point: $( 4 , - 2 , - 4 )$

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

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. $\mathbf { u } = \langle - 3,6,3 \rangle , \mathbf { v } = \langle 9,9,9 \rangle$

Find the area of the parallelogram formed by the points $A ( 2 , - 3,5 ) , B ( 7 , - 2,7 )$ , $C ( 3 , - 2,11 )$ , and $D ( 8 , - 1,13 )$ .

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

Find a set of parametric equations for the line through the point and parallel to the specified vector. Show all your work. $( - 1 , - 5,9 )$ , parallel to $\langle - 7,6 , - 3 \rangle$

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 general form of the equation of the plane passing through the three points. [Be sure to reduce the coefficients in your answer to lowest terms by dividing out any common factor.] $( 5 , - 5 , - 6 ) , ( 6 , - 1,4 ) , ( - 2 , - 5,5 )$

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 a set of parametric equations for the line that passes through the given points. Show all your work. $\left( 4 , \frac { 9 } { 2 } , \frac { - 3 } { 2 } \right) , \left( \frac { 1 } { 2 } , \frac { - 3 } { 2 } , - 2 \right)$

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. $\mathbf { u } = - 6 \mathbf { i } + 9 \mathbf { j } + 2 \mathbf { k } , \mathbf { v } = - 3 \mathbf { i } - 7 \mathbf { j } - 4 \mathbf { k }$

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

Functions and Their Graphs

Polynomial and Rational Functions

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Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

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Sequences, Series, and Probability

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Calculus Practice Problems

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Exam 10: Analytic Geometry in Three Dimensions

Find the midpoint of the line segment joining the points. (−5,8,9),(−8,9,2)( - 5,8,9 ) , ( - 8,9,2 )(−5,8,9),(−8,9,2)

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

Find the vector z\mathbf { z }z , given u=⟨−3,3,9⟩\mathbf { u } = \langle - 3,3,9 \rangleu=⟨−3,3,9⟩ and v=⟨−7,5,7⟩\mathbf { v } = \langle - 7,5,7 \ranglev=⟨−7,5,7⟩ . z=−3u−5v\mathbf { z } = - 3 \mathbf { u } - 5 \mathbf { v }z=−3u−5v

Find the magnitude of the vector v\mathbf { v }v . v=⟨0,−5,−3⟩\mathbf { v } = \langle 0 , - 5 , - 3 \ranglev=⟨0,−5,−3⟩

Find a set of parametric equations for the line that passes through the given points. Show all your work. (4,−7,2),(−7,−1,−5)( 4 , - 7,2 ) , ( - 7 , - 1 , - 5 )(4,−7,2),(−7,−1,−5)

Write the component form of the vector described below. Initial point: (−1,−5,−6)( - 1 , - 5 , - 6 )(−1,−5,−6) Terminal point: (4,−2,−4)( 4 , - 2 , - 4 )(4,−2,−4)

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

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⟩

Find the area of the parallelogram formed by the points A(2,−3,5),B(7,−2,7)A ( 2 , - 3,5 ) , B ( 7 , - 2,7 )A(2,−3,5),B(7,−2,7) , C(3,−2,11)C ( 3 , - 2,11 )C(3,−2,11) , and D(8,−1,13)D ( 8 , - 1,13 )D(8,−1,13) .

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 a set of parametric equations for the line through the point and parallel to the specified vector. Show all your work. (−1,−5,9)( - 1 , - 5,9 )(−1,−5,9) , parallel to ⟨−7,6,−3⟩\langle - 7,6 , - 3 \rangle⟨−7,6,−3⟩

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 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 a set of parametric equations for the line that passes through the given points. Show all your work. (4,92,−32),(12,−32,−2)\left( 4 , \frac { 9 } { 2 } , \frac { - 3 } { 2 } \right) , \left( \frac { 1 } { 2 } , \frac { - 3 } { 2 } , - 2 \right)(4,29​,2−3​),(21​,2−3​,−2)

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

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

Calculus Practice Problems

Filters

Find the midpoint of the line segment joining the points. $( - 5,8,9 ) , ( - 8,9,2 )$

Find the vector $\mathbf { z }$ , given $\mathbf { u } = \langle - 3,3,9 \rangle$ and $\mathbf { v } = \langle - 7,5,7 \rangle$ . $\mathbf { z } = - 3 \mathbf { u } - 5 \mathbf { v }$

Find the magnitude of the vector $\mathbf { v }$ . $\mathbf { v } = \langle 0 , - 5 , - 3 \rangle$

Find a set of parametric equations for the line that passes through the given points. Show all your work. $( 4 , - 7,2 ) , ( - 7 , - 1 , - 5 )$

Write the component form of the vector described below. Initial point: $( - 1 , - 5 , - 6 )$ Terminal point: $( 4 , - 2 , - 4 )$

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. $\mathbf { u } = \langle - 3,6,3 \rangle , \mathbf { v } = \langle 9,9,9 \rangle$

Find the area of the parallelogram formed by the points $A ( 2 , - 3,5 ) , B ( 7 , - 2,7 )$ , $C ( 3 , - 2,11 )$ , and $D ( 8 , - 1,13 )$ .

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

Find a set of parametric equations for the line through the point and parallel to the specified vector. Show all your work. $( - 1 , - 5,9 )$ , parallel to $\langle - 7,6 , - 3 \rangle$

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 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 a set of parametric equations for the line that passes through the given points. Show all your work. $\left( 4 , \frac { 9 } { 2 } , \frac { - 3 } { 2 } \right) , \left( \frac { 1 } { 2 } , \frac { - 3 } { 2 } , - 2 \right)$

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