Question 1

Find the distance between the points.
$$( 1,5,5 ) , ( - 2,6 , - 1 )$$

Accepted Answer

A)  14 
B)  10 
C)  $\sqrt { 10 }$ 
D)  $2 \sqrt { 46 }$ 
E)  $\sqrt { 46 }$ 
A)  14 
B)  10 
C)  $\sqrt { 10 }$ 
D)  $2 \sqrt { 46 }$ 
E)  $\sqrt { 46 }$

Question 2

Find the general form of the equation of the plane passing through the point and perpendicular to the specified line.
$x = 2 + 3 t$
$( 5 , - 6 , - 4 ) , \quad y = - 3 + 4 t$
$z = - 6 + 6 t$

Accepted Answer

A)  $3 x + 4 y + 6 z = 0$ 
B)  $3 x + 4 y + 6 z - 33 = 0$ 
C)  $3 x + 4 y + 6 z + 33 = 0$ 
D)  $5 x - 6 y - 4 z - 33 = 0$ 
E)  $5 x - 6 y - 4 z + 33 = 0$ 
A)  $3 x + 4 y + 6 z = 0$ 
B)  $3 x + 4 y + 6 z - 33 = 0$ 
C)  $3 x + 4 y + 6 z + 33 = 0$ 
D)  $5 x - 6 y - 4 z - 33 = 0$ 
E)  $5 x - 6 y - 4 z + 33 = 0$

Question 3

Find symmetric equations for the line through the point and parallel to the specified vector. Show all your work.
$( 1,8,6 )$, parallel to $\langle 9 , - 6,5 \rangle$

Accepted Answer

Answers may vary. On

Question 4

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(d) $\mathbf { i } \operatorname { coeff } ( \mathrm { f } ) \mathbf { j } \operatorname { coeff } ( \mathrm { g } ) \mathbf { k }$

Accepted Answer

A) 
$\frac { 1 } { \text { leadcoeff( } ( d a 1 \mathrm { fac } ) \sqrt { \text { da 1 rad } } } \left( \right.$ leadcoeff( $\left. \left. \mathrm { a } ^ { * } \mathrm {~d} \right) \mathbf { i } \operatorname { coeff } \left( \mathrm { b } ^ { * } f \right) \mathbf { j } \operatorname { coeff } \left( \mathrm { c } ^ { * } g \right) \mathbf { k } \right)$ 
B) 
$\frac { 1 } { \text { leadcoeff } ( d a 2 f a c ) \sqrt { d a 2 r a d } } \left( \right.$ 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) 
$\frac { 1 } { \text { leadcoeff( } ( d a 1 \mathrm { fac } ) \sqrt { \text { da 1 rad } } } \left( \right.$ leadcoeff( $\left. \left. \mathrm { a } ^ { * } \mathrm {~d} \right) \mathbf { i } \operatorname { coeff } \left( \mathrm { b } ^ { * } f \right) \mathbf { j } \operatorname { coeff } \left( \mathrm { c } ^ { * } g \right) \mathbf { k } \right)$ 
B) 
$\frac { 1 } { \text { leadcoeff } ( d a 2 f a c ) \sqrt { d a 2 r a d } } \left( \right.$ 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 5

Find the dot product of $\mathbf { u }$ and $\mathbf { v }$.
$$\mathbf { u } = \langle - 3 , - 2,4 \rangle , \mathbf { v } = \langle - 4 , - 5 , - 7 \rangle$$

Accepted Answer

A)  $- 6$ 
B)  $- 17$ 
C)  $\langle 12,10 , - 28 \rangle$ 
D)  50 
E)  $\langle - 7 , - 7 , - 3 \rangle$ 
A)  $- 6$ 
B)  $- 17$ 
C)  $\langle 12,10 , - 28 \rangle$ 
D)  50 
E)  $\langle - 7 , - 7 , - 3 \rangle$

Question 6

Find the distance between the point and the plane.
$( - 2 , - 1 , - 4 )$
$- 3 x - 2 y - z = 1$

Accepted Answer

A)  11 
B)  $\frac { 11 } { 14 }$ 
C)  $\frac { 13 } { \sqrt { 14 } }$ 
D)  0 
E)  $\frac { 11 } { \sqrt { 14 } }$ 
A)  11 
B)  $\frac { 11 } { 14 }$ 
C)  $\frac { 13 } { \sqrt { 14 } }$ 
D)  0 
E)  $\frac { 11 } { \sqrt { 14 } }$

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

Find the volume of the parallelpiped with the given vertices. $A ( 0 , - 5,3 ) , B ( 3 , - 1 , - 4 ) , C ( 7 , - 3 , - 1 ) , D ( 10,1 , - 8 )$, $E ( - 3 , - 13,6 ) , F ( 0 , - 9 , - 1 ) , G ( 4 , - 11,2 ) , H ( 7 , - 7 , - 5 )$

Accepted Answer

A)  43 
B)  236 
C)  308 
D)  296 
E)  256 
A)  43 
B)  236 
C)  308 
D)  296 
E)  256

Question 9

Find the area of the parallelogram that has the vectors as adjacent sides.
$$\mathbf { u } = \langle 4 , - 1,4 \rangle , \mathbf { v } = \langle 2 , - 3,0 \rangle$$

Accepted Answer

A)  11 
B)  $\sqrt { 77 }$ 
C)  $\sqrt { 11 }$ 
D)  4 
E)  $2 \sqrt { 77 }$ 
A)  11 
B)  $\sqrt { 77 }$ 
C)  $\sqrt { 11 }$ 
D)  4 
E)  $2 \sqrt { 77 }$

Question 10

Find the midpoint of the line segment joining the points.
$$( - 2 , - 7,3 ) , ( - 3 , - 4 , - 6 )$$

Accepted Answer

A)  $\left( \frac { 1 } { 2 } , \frac { - 3 } { 2 } , \frac { 9 } { 2 } \right)$ 
B)  $\left( \frac { - 5 } { 2 } , \frac { - 11 } { 2 } , \frac { - 3 } { 2 } \right)$ 
C)  $( - 5 , - 11 , - 3 )$ 
D)  $( 1 , - 3,9 )$ 
E)  $( 6,28 , - 18 )$ 
A)  $\left( \frac { 1 } { 2 } , \frac { - 3 } { 2 } , \frac { 9 } { 2 } \right)$ 
B)  $\left( \frac { - 5 } { 2 } , \frac { - 11 } { 2 } , \frac { - 3 } { 2 } \right)$ 
C)  $( - 5 , - 11 , - 3 )$ 
D)  $( 1 , - 3,9 )$ 
E)  $( 6,28 , - 18 )$

Question 11

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 12

Find the standard form of the equation of the sphere with the given characteristics.
Endpoints of a diameter: $( - 8,4 , - 2 ) , ( - 8 , - 2,6 )$

Accepted Answer

A)  $( x + 16 ) ^ { 2 } + ( x - 2 ) ^ { 2 } + ( x - 4 ) ^ { 2 } = 25$ 
B)  $( x + 8 ) ^ { 2 } + ( x - 1 ) ^ { 2 } + ( x - 2 ) ^ { 2 } = 50$ 
C)  $( x + 8 ) ^ { 2 } + ( x - 1 ) ^ { 2 } + ( x - 2 ) ^ { 2 } = 100$ 
D)  $( x + 8 ) ^ { 2 } + ( x - 1 ) ^ { 2 } + ( x - 2 ) ^ { 2 } = 25$ 
E)  $( x - 8 ) ^ { 2 } + ( x + 1 ) ^ { 2 } + ( x + 2 ) ^ { 2 } = 25$ 
A)  $( x + 16 ) ^ { 2 } + ( x - 2 ) ^ { 2 } + ( x - 4 ) ^ { 2 } = 25$ 
B)  $( x + 8 ) ^ { 2 } + ( x - 1 ) ^ { 2 } + ( x - 2 ) ^ { 2 } = 50$ 
C)  $( x + 8 ) ^ { 2 } + ( x - 1 ) ^ { 2 } + ( x - 2 ) ^ { 2 } = 100$ 
D)  $( x + 8 ) ^ { 2 } + ( x - 1 ) ^ { 2 } + ( x - 2 ) ^ { 2 } = 25$ 
E)  $( x - 8 ) ^ { 2 } + ( x + 1 ) ^ { 2 } + ( x + 2 ) ^ { 2 } = 25$

Question 13

Find a set of symmetric equations of the line that passes through the points $( 5,0,5 )$ and $( 7,8 , - 6 )$.

Accepted Answer

A)  $\frac { x } { 7 } = \frac { y } { 8 } = \frac { z } { - 6 }$ 
B)  $\frac { x - 5 } { 2 } = \frac { y - 8 } { 8 } = \frac { z - 6 } { - 11 }$ 
C)  $\frac { x - 5 } { - 2 } = \frac { y } { 8 } = \frac { z - 5 } { 11 }$ 
D)  $\frac { x + 2 } { 5 } = y - 8 = \frac { z - 11 } { 5 }$ 
E)  $\frac { x - 5 } { 2 } = \frac { y } { 8 } = \frac { z - 5 } { - 11 }$ 
A)  $\frac { x } { 7 } = \frac { y } { 8 } = \frac { z } { - 6 }$ 
B)  $\frac { x - 5 } { 2 } = \frac { y - 8 } { 8 } = \frac { z - 6 } { - 11 }$ 
C)  $\frac { x - 5 } { - 2 } = \frac { y } { 8 } = \frac { z - 5 } { 11 }$ 
D)  $\frac { x + 2 } { 5 } = y - 8 = \frac { z - 11 } { 5 }$ 
E)  $\frac { x - 5 } { 2 } = \frac { y } { 8 } = \frac { z - 5 } { - 11 }$

Question 14

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are parallel, orthogonal, or neither.
$$\mathbf { u } = \langle 6 , - 5,2 \rangle , \mathbf { v } = \langle 30 , - 25,10 \rangle$$

Accepted Answer

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

Question 15

The weight of a crate is 300 newtons. Find the tension in each of the supporting cables shown in the figure. The coordinates of the points $A , B , C$, and $D$ are given below the figure. Round to the nearest newton.

[Figure not necessarily to scale.]
point $A = ( 0,0 , - 130 )$, point $B = ( 90,0,0 )$, point $C = ( - 40,40,0 )$, point $D = ( 0 , - 180,0 )$

Accepted Answer

A)  cable $A B = 196$; cable $A C = 97$; cable $A D = 68$ 
B)  cable $A B = 97$; cable $A C = 68$; cable $A D = 196$ 
C)  cable $A B = 97$; cable $A C = 196$; cable $A D = 68$ 
D)  cable $A B = 68$; cable $A C = 196$; cable $A D = 97$ 
E)  cable $A B = 196$; cable $A C = 68 ;$ cable $A D = 97$ ) 
A)  cable $A B = 196$; cable $A C = 97$; cable $A D = 68$ 
B)  cable $A B = 97$; cable $A C = 68$; cable $A D = 196$ 
C)  cable $A B = 97$; cable $A C = 196$; cable $A D = 68$ 
D)  cable $A B = 68$; cable $A C = 196$; cable $A D = 97$ 
E)  cable $A B = 196$; cable $A C = 68 ;$ cable $A D = 97$ )

Question 16

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)  $\quad c = \pm \frac { 1 } { 6 }$ 
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)  $\quad c = \pm \frac { 1 } { 6 }$ 
E)  $c = \pm \frac { \sqrt { 86 } } { 86 }$

Question 17

Find symmetric equations for the line through the point and parallel to the specified line.
Show all your work. $x=-1-3 t$
$(-6,-2,5) \text {, parallel to } y=6-8 t$
$z=4-3 t$

Accepted Answer

Answers may vary. On

Question 18

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 19

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 20

Find symmetric equations for the line through the point and parallel to the specified line. Show all your work. 
$x = - 8 - 9 t$ 
$(5,-9,-1) \text {, parallel to } y=5-5 t$
z = - 8 + 3 t

Accepted Answer

Answers may vary. On

Find the distance between the points. $( 1,5,5 ) , ( - 2,6 , - 1 )$

Find the general form of the equation of the plane passing through the point and perpendicular to the specified line. $x = 2 + 3 t$ $( 5 , - 6 , - 4 ) , \quad y = - 3 + 4 t$ $z = - 6 + 6 t$

Find symmetric equations for the line through the point and parallel to the specified vector. Show all your work. $( 1,8,6 )$ , parallel to $\langle 9 , - 6,5 \rangle$

Find the dot product of $\mathbf { u }$ and $\mathbf { v }$ . $\mathbf { u } = \langle - 3 , - 2,4 \rangle , \mathbf { v } = \langle - 4 , - 5 , - 7 \rangle$

Find the distance between the point and the plane. $( - 2 , - 1 , - 4 )$ $- 3 x - 2 y - z = 1$

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

Find the volume of the parallelpiped with the given vertices. $A ( 0 , - 5,3 ) , B ( 3 , - 1 , - 4 ) , C ( 7 , - 3 , - 1 ) , D ( 10,1 , - 8 )$ , $E ( - 3 , - 13,6 ) , F ( 0 , - 9 , - 1 ) , G ( 4 , - 11,2 ) , H ( 7 , - 7 , - 5 )$

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = \langle 4 , - 1,4 \rangle , \mathbf { v } = \langle 2 , - 3,0 \rangle$

Find the midpoint of the line segment joining the points. $( - 2 , - 7,3 ) , ( - 3 , - 4 , - 6 )$

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

Find the standard form of the equation of the sphere with the given characteristics. Endpoints of a diameter: $( - 8,4 , - 2 ) , ( - 8 , - 2,6 )$

Find a set of symmetric equations of the line that passes through the points $( 5,0,5 )$ and $( 7,8 , - 6 )$ .

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are parallel, orthogonal, or neither. $\mathbf { u } = \langle 6 , - 5,2 \rangle , \mathbf { v } = \langle 30 , - 25,10 \rangle$

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. $x=-1-3 t$ $(-6,-2,5) \text {, parallel to } y=6-8 t$ $z=4-3 t$

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

Find symmetric equations for the line through the point and parallel to the specified line. Show all your work. $x = - 8 - 9 t$ $(5,-9,-1) \text {, parallel to } y=5-5 t$ z = - 8 + 3 t

Functions and Their Graphs

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Exam 10: Analytic Geometry in Three Dimensions

Find the distance between the points. (1,5,5),(−2,6,−1)( 1,5,5 ) , ( - 2,6 , - 1 )(1,5,5),(−2,6,−1)

Find the general form of the equation of the plane passing through the point and perpendicular to the specified line. x=2+3tx = 2 + 3 tx=2+3t (5,−6,−4),y=−3+4t( 5 , - 6 , - 4 ) , \quad y = - 3 + 4 t(5,−6,−4),y=−3+4t z=−6+6tz = - 6 + 6 tz=−6+6t

Find symmetric equations for the line through the point and parallel to the specified vector. Show all your work. (1,8,6)( 1,8,6 )(1,8,6) , parallel to ⟨9,−6,5⟩\langle 9 , - 6,5 \rangle⟨9,−6,5⟩

Find the dot product of u\mathbf { u }u and v\mathbf { v }v . u=⟨−3,−2,4⟩,v=⟨−4,−5,−7⟩\mathbf { u } = \langle - 3 , - 2,4 \rangle , \mathbf { v } = \langle - 4 , - 5 , - 7 \rangleu=⟨−3,−2,4⟩,v=⟨−4,−5,−7⟩

Find the distance between the point and the plane. (−2,−1,−4)( - 2 , - 1 , - 4 )(−2,−1,−4) −3x−2y−z=1- 3 x - 2 y - z = 1−3x−2y−z=1

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)

Find the area of the parallelogram that has the vectors as adjacent sides. u=⟨4,−1,4⟩,v=⟨2,−3,0⟩\mathbf { u } = \langle 4 , - 1,4 \rangle , \mathbf { v } = \langle 2 , - 3,0 \rangleu=⟨4,−1,4⟩,v=⟨2,−3,0⟩

Find the midpoint of the line segment joining the points. (−2,−7,3),(−3,−4,−6)( - 2 , - 7,3 ) , ( - 3 , - 4 , - 6 )(−2,−7,3),(−3,−4,−6)

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 standard form of the equation of the sphere with the given characteristics. Endpoints of a diameter: (−8,4,−2),(−8,−2,6)( - 8,4 , - 2 ) , ( - 8 , - 2,6 )(−8,4,−2),(−8,−2,6)

Find a set of symmetric equations of the line that passes through the points (5,0,5)( 5,0,5 )(5,0,5) and (7,8,−6)( 7,8 , - 6 )(7,8,−6) .

Determine whether u\mathbf { u }u and v\mathbf { v }v are parallel, orthogonal, or neither. u=⟨6,−5,2⟩,v=⟨30,−25,10⟩\mathbf { u } = \langle 6 , - 5,2 \rangle , \mathbf { v } = \langle 30 , - 25,10 \rangleu=⟨6,−5,2⟩,v=⟨30,−25,10⟩

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=−1−3tx=-1-3 tx=−1−3t (−6,−2,5), parallel to y=6−8t(-6,-2,5) \text {, parallel to } y=6-8 t(−6,−2,5), parallel to y=6−8t z=4−3tz=4-3 tz=4−3t

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⟩

Find symmetric equations for the line through the point and parallel to the specified line. Show all your work. x=−8−9tx = - 8 - 9 tx=−8−9t (5,−9,−1), parallel to y=5−5t(5,-9,-1) \text {, parallel to } y=5-5 t(5,−9,−1), parallel to y=5−5t z = - 8 + 3 t

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

Review of Graphs, Equations, and Inequalities

Filters

Find the distance between the points. $( 1,5,5 ) , ( - 2,6 , - 1 )$

Find the general form of the equation of the plane passing through the point and perpendicular to the specified line. $x = 2 + 3 t$ $( 5 , - 6 , - 4 ) , \quad y = - 3 + 4 t$ $z = - 6 + 6 t$

Find symmetric equations for the line through the point and parallel to the specified vector. Show all your work. $( 1,8,6 )$ , parallel to $\langle 9 , - 6,5 \rangle$

Find the dot product of $\mathbf { u }$ and $\mathbf { v }$ . $\mathbf { u } = \langle - 3 , - 2,4 \rangle , \mathbf { v } = \langle - 4 , - 5 , - 7 \rangle$

Find the distance between the point and the plane. $( - 2 , - 1 , - 4 )$ $- 3 x - 2 y - z = 1$

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

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = \langle 4 , - 1,4 \rangle , \mathbf { v } = \langle 2 , - 3,0 \rangle$

Find the midpoint of the line segment joining the points. $( - 2 , - 7,3 ) , ( - 3 , - 4 , - 6 )$

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

Find the standard form of the equation of the sphere with the given characteristics. Endpoints of a diameter: $( - 8,4 , - 2 ) , ( - 8 , - 2,6 )$

Find a set of symmetric equations of the line that passes through the points $( 5,0,5 )$ and $( 7,8 , - 6 )$ .

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are parallel, orthogonal, or neither. $\mathbf { u } = \langle 6 , - 5,2 \rangle , \mathbf { v } = \langle 30 , - 25,10 \rangle$

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. $x=-1-3 t$ $(-6,-2,5) \text {, parallel to } y=6-8 t$ $z=4-3 t$

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

Find symmetric equations for the line through the point and parallel to the specified line. Show all your work. $x = - 8 - 9 t$ $(5,-9,-1) \text {, parallel to } y=5-5 t$ z = - 8 + 3 t