Question 1

Find the angle of intersection of the planes in degrees. Round to a tenth of a degree.
$$\begin{array} { l } 
3 x - 2 y + 5 z = 3 \
- x + 3 y + 3 z = 3
\end{array}$$

Accepted Answer

A)  $4.8 ^ { \circ }$ 
B)  $12.9 ^ { \circ }$ 
C)  $77.1 ^ { \circ }$ 
D)  $12.6 ^ { \circ }$ 
E)  $1.3 ^ { \circ }$ 
A)  $4.8 ^ { \circ }$ 
B)  $12.9 ^ { \circ }$ 
C)  $77.1 ^ { \circ }$ 
D)  $12.6 ^ { \circ }$ 
E)  $1.3 ^ { \circ }$

Question 2

Find the acute interior angle of the parallelogram formed by $A ( 2 , - 2,2 ) , B ( 5 , - 1,4 )$, $C ( 6,0,10 )$, and $D ( 3 , - 1,8 )$. Round your answer to two decimals.

Accepted Answer

A)  $43.92 ^ { \circ }$ 
B)  $35.76 ^ { \circ }$ 
C)  $46.08 ^ { \circ }$ 
D)  $1.79$ 
E)  $31.25 ^ { \circ }$ 
A)  $43.92 ^ { \circ }$ 
B)  $35.76 ^ { \circ }$ 
C)  $46.08 ^ { \circ }$ 
D)  $1.79$ 
E)  $31.25 ^ { \circ }$

Question 3

Find $\mathbf { u } \times \mathbf { v }$.
$$\mathbf { u } = - 8 \mathbf { i } - 5 \mathbf { j } + 3 \mathbf { k } , \mathbf { v } = - 6 \mathbf { i } - 7 \mathbf { j } - 3 \mathbf { k }$$

Accepted Answer

A)  $36 \mathbf { i } - 42 \mathbf { j } + 26 \mathbf { k }$ 
B)  74 
C)  4 
D)  $36 \mathbf { i } + 42 \mathbf { j } + 26 \mathbf { k }$ 
E)  $48 \mathbf { i } + 35 \mathbf { j } - 9 \mathbf { k }$ 
A)  $36 \mathbf { i } - 42 \mathbf { j } + 26 \mathbf { k }$ 
B)  74 
C)  4 
D)  $36 \mathbf { i } + 42 \mathbf { j } + 26 \mathbf { k }$ 
E)  $48 \mathbf { i } + 35 \mathbf { j } - 9 \mathbf { k }$

Question 4

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.]
$$( - 2 , - 1,4 ) , ( 3 , - 1,2 ) , ( - 6 , - 5 , - 6 )$$

Accepted Answer

A)  $4 x - 29 y + 10 z - 61 = 0$ 
B)  $4 x - 29 y + 10 z = 0$ 
C)  $2 x + y - 4 z = 0$ 
D)  $4 x - 29 y + 10 z + 61 = 0$ 
E)  $\quad 2 x + y - 4 z + 122 = 0$ 
A)  $4 x - 29 y + 10 z - 61 = 0$ 
B)  $4 x - 29 y + 10 z = 0$ 
C)  $2 x + y - 4 z = 0$ 
D)  $4 x - 29 y + 10 z + 61 = 0$ 
E)  $\quad 2 x + y - 4 z + 122 = 0$

Question 5

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 6

Find the lengths of the sides of the right triangle whose vertices are located at the given points. Show that these lengths satisfy the Pythagorean Theorem. Show all of your
work.
(8, -4,4) ,(9,5,3),(-3,6,0)

Accepted Answer

Let point @#LAT-DLM&, point @#LAT-DLM&, and point @#LAT-DLM&.
\[\begin

Question 7

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 8

Find the magnitude of the vector $\mathbf { v }$.
$$\mathbf { v } = \langle 7 , - 4,0 \rangle$$

Accepted Answer

A)  3 
B)  11 
C)  $\sqrt { 3 }$ 
D)  $2 \sqrt { 65 }$ 
E)  $\sqrt { 65 }$ 
A)  3 
B)  11 
C)  $\sqrt { 3 }$ 
D)  $2 \sqrt { 65 }$ 
E)  $\sqrt { 65 }$

Question 9

Find the angle, in degrees, between two adjacent sides of the pyramid shown below. Round to the nearest tenth of a degree. [Note: The base of the pyramid is not considered a side.]

Accepted Answer

A)  $2.3 ^ { \circ }$ 
B)  $129.8 ^ { \circ }$ 
C)  $90 ^ { \circ }$ 
D)  $45 ^ { \circ }$ 
E)  $39.8 ^ { \circ }$ 
A)  $2.3 ^ { \circ }$ 
B)  $129.8 ^ { \circ }$ 
C)  $90 ^ { \circ }$ 
D)  $45 ^ { \circ }$ 
E)  $39.8 ^ { \circ }$

Question 10

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 11

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

Accepted Answer

A)  $\langle 18 , - 6 , - 9 \rangle$ 
B)  $\langle 48,9 , - 39 \rangle$ 
C)  $\langle - 14 , - 2,11 \rangle$ 
D)  $\langle 50,5 , - 38 \rangle$ 
E)  $\langle 0 , - 15,9 \rangle$ 
A)  $\langle 18 , - 6 , - 9 \rangle$ 
B)  $\langle 48,9 , - 39 \rangle$ 
C)  $\langle - 14 , - 2,11 \rangle$ 
D)  $\langle 50,5 , - 38 \rangle$ 
E)  $\langle 0 , - 15,9 \rangle$

Question 12

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 13

Find the midpoint of the line segment joining the points.
$$( - 3,7 , - 8 ) , ( 8,2,5 )$$

Accepted Answer

A)  $\left( \frac { - 11 } { 2 } , \frac { 5 } { 2 } , \frac { - 13 } { 2 } \right)$ 
B)  $\left( \frac { 5 } { 2 } , \frac { 9 } { 2 } , \frac { - 3 } { 2 } \right)$ 
C)  $( 5,9 , - 3 )$ 
D)  $( - 11,5 , - 13 )$ 
E)  $( - 24,14 , - 40 )$ 
A)  $\left( \frac { - 11 } { 2 } , \frac { 5 } { 2 } , \frac { - 13 } { 2 } \right)$ 
B)  $\left( \frac { 5 } { 2 } , \frac { 9 } { 2 } , \frac { - 3 } { 2 } \right)$ 
C)  $( 5,9 , - 3 )$ 
D)  $( - 11,5 , - 13 )$ 
E)  $( - 24,14 , - 40 )$

Question 14

Find the acute interior angle of the parallelogram formed by $A ( 1 , - 1,2 ) , B ( 6,0,4 )$, $C ( 7,1,9 )$, and $D ( 2,0,7 )$. Round your answer to two decimals.

Accepted Answer

A)  $34.21 ^ { \circ }$ 
B)  $39.59 ^ { \circ }$ 
C)  $55.79 ^ { \circ }$ 
D)  $1.67 ^ { \circ }$ 
E)  $43.15 ^ { \circ }$ 
A)  $34.21 ^ { \circ }$ 
B)  $39.59 ^ { \circ }$ 
C)  $55.79 ^ { \circ }$ 
D)  $1.67 ^ { \circ }$ 
E)  $43.15 ^ { \circ }$

Question 15

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

Accepted Answer

Answers may vary. On

Question 16

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)  $\langle - 26,16,8 \rangle$

Question 17

$\text { For the points } A ( 4 , - 1 , - 1 ) , B ( - 1 , - 2,4 ) , C ( 5 , - 4,4 ) , D ( 0 ,- 5,9) \text { : }$ 
a. Verify that the points are vertices of a parallelogram. Show all work.
b. Find the area of the parallelogram. Show all work.
c. Determine whether the parallelogram is a rectangle.

Accepted Answer

Answer details may vary.
a.
@#LAT-DLM& \overrighta

Question 18

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.
$$\mathbf { u } = - 5 \mathbf { i } - 6 \mathbf { j } - \mathbf { k } , \mathbf { v } = 6 \mathbf { i } + \mathbf { j } + \mathbf { k }$$

Accepted Answer

A)  $37.3 ^ { \circ }$ 
B)  $49.7 ^ { \circ }$ 
C)  $52.7 ^ { \circ }$ 
D)  $90 ^ { \circ }$ 
E)  $139.7 ^ { \circ }$ 
A)  $37.3 ^ { \circ }$ 
B)  $49.7 ^ { \circ }$ 
C)  $52.7 ^ { \circ }$ 
D)  $90 ^ { \circ }$ 
E)  $139.7 ^ { \circ }$

Question 19

Find the angle of intersection of the planes in degrees. Round to a tenth of a degree.
$$\begin{array} { l } 
3 x - 2 y + 4 z = 3 \
5 x - 3 y + 3 z = - 6
\end{array}$$

Accepted Answer

A)  $- 0.8 ^ { \circ }$ 
B)  $69.1 ^ { \circ }$ 
C)  $20.9 ^ { \circ }$ 
D)  $43.1 ^ { \circ }$ 
E)  $0.4 ^ { \circ }$ 
A)  $- 0.8 ^ { \circ }$ 
B)  $69.1 ^ { \circ }$ 
C)  $20.9 ^ { \circ }$ 
D)  $43.1 ^ { \circ }$ 
E)  $0.4 ^ { \circ }$

Question 20

Find the magnitude of the vector described below.
Initial point: $( 5,6 , - 4 )$
Terminal point: $( - 4,2,7 )$

Accepted Answer

A)  2 
B)  36 
C)  $2 \sqrt { 6 }$ 
D)  $2 \sqrt { 218 }$ 
E)  $\sqrt { 218 }$ 
A)  2 
B)  36 
C)  $2 \sqrt { 6 }$ 
D)  $2 \sqrt { 218 }$ 
E)  $\sqrt { 218 }$

Find the angle of intersection of the planes in degrees. Round to a tenth of a degree. 3x-2y+5z=3 -x+3y+3z=3

Find the acute interior angle of the parallelogram formed by $A ( 2 , - 2,2 ) , B ( 5 , - 1,4 )$ , $C ( 6,0,10 )$ , and $D ( 3 , - 1,8 )$ . Round your answer to two decimals.

Find $\mathbf { u } \times \mathbf { v }$ . $\mathbf { u } = - 8 \mathbf { i } - 5 \mathbf { j } + 3 \mathbf { k } , \mathbf { v } = - 6 \mathbf { i } - 7 \mathbf { j } - 3 \mathbf { k }$

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.] $( - 2 , - 1,4 ) , ( 3 , - 1,2 ) , ( - 6 , - 5 , - 6 )$

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 lengths of the sides of the right triangle whose vertices are located at the given points. Show that these lengths satisfy the Pythagorean Theorem. Show all of your work. (8, -4,4) ,(9,5,3),(-3,6,0)

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

Find the magnitude of the vector $\mathbf { v }$ . $\mathbf { v } = \langle 7 , - 4,0 \rangle$

Find the angle, in degrees, between two adjacent sides of the pyramid shown below. Round to the nearest tenth of a degree. [Note: The base of the pyramid is not considered a side.]

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 vector $\mathbf { z }$ , given $\mathbf { u } = \langle - 8,1,5 \rangle$ and $\mathbf { v } = \langle - 6 , - 3,6 \rangle$ . $\mathbf { z } = - 3 \mathbf { u } - 4 \mathbf { v }$

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

Find the midpoint of the line segment joining the points. $( - 3,7 , - 8 ) , ( 8,2,5 )$

Find the acute interior angle of the parallelogram formed by $A ( 1 , - 1,2 ) , B ( 6,0,4 )$ , $C ( 7,1,9 )$ , and $D ( 2,0,7 )$ . Round your answer to two decimals.

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

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

$\text { For the points } A ( 4 , - 1 , - 1 ) , B ( - 1 , - 2,4 ) , C ( 5 , - 4,4 ) , D ( 0 ,- 5,9) \text { : }$ a. Verify that the points are vertices of a parallelogram. Show all work. b. Find the area of the parallelogram. Show all work. c. Determine whether the parallelogram is a rectangle.

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. $\mathbf { u } = - 5 \mathbf { i } - 6 \mathbf { j } - \mathbf { k } , \mathbf { v } = 6 \mathbf { i } + \mathbf { j } + \mathbf { k }$

Find the angle of intersection of the planes in degrees. Round to a tenth of a degree. 3x-2y+4z=3 5x-3y+3z=-6

Find the magnitude of the vector described below. Initial point: $( 5,6 , - 4 )$ Terminal point: $( - 4,2,7 )$

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Exam 10: Analytic Geometry in Three Dimensions

Find the angle of intersection of the planes in degrees. Round to a tenth of a degree. 3x-2y+5z=3 -x+3y+3z=3

Find the acute interior angle of the parallelogram formed by A(2,−2,2),B(5,−1,4)A ( 2 , - 2,2 ) , B ( 5 , - 1,4 )A(2,−2,2),B(5,−1,4) , C(6,0,10)C ( 6,0,10 )C(6,0,10) , and D(3,−1,8)D ( 3 , - 1,8 )D(3,−1,8) . Round your answer to two decimals.

Find u×v\mathbf { u } \times \mathbf { v }u×v . u=−8i−5j+3k,v=−6i−7j−3k\mathbf { u } = - 8 \mathbf { i } - 5 \mathbf { j } + 3 \mathbf { k } , \mathbf { v } = - 6 \mathbf { i } - 7 \mathbf { j } - 3 \mathbf { k }u=−8i−5j+3k,v=−6i−7j−3k

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 lengths of the sides of the right triangle whose vertices are located at the given points. Show that these lengths satisfy the Pythagorean Theorem. Show all of your work. (8, -4,4) ,(9,5,3),(-3,6,0)

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

Find the magnitude of the vector v\mathbf { v }v . v=⟨7,−4,0⟩\mathbf { v } = \langle 7 , - 4,0 \ranglev=⟨7,−4,0⟩

Find the angle, in degrees, between two adjacent sides of the pyramid shown below. Round to the nearest tenth of a degree. [Note: The base of the pyramid is not considered a side.]

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 vector z\mathbf { z }z , given u=⟨−8,1,5⟩\mathbf { u } = \langle - 8,1,5 \rangleu=⟨−8,1,5⟩ and v=⟨−6,−3,6⟩\mathbf { v } = \langle - 6 , - 3,6 \ranglev=⟨−6,−3,6⟩ . z=−3u−4v\mathbf { z } = - 3 \mathbf { u } - 4 \mathbf { v }z=−3u−4v

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

Find the midpoint of the line segment joining the points. (−3,7,−8),(8,2,5)( - 3,7 , - 8 ) , ( 8,2,5 )(−3,7,−8),(8,2,5)

Find the acute interior angle of the parallelogram formed by A(1,−1,2),B(6,0,4)A ( 1 , - 1,2 ) , B ( 6,0,4 )A(1,−1,2),B(6,0,4) , C(7,1,9)C ( 7,1,9 )C(7,1,9) , and D(2,0,7)D ( 2,0,7 )D(2,0,7) . Round your answer to two decimals.

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

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 angle of intersection of the planes in degrees. Round to a tenth of a degree. 3x-2y+4z=3 5x-3y+3z=-6

Find the magnitude of the vector described below. Initial point: (5,6,−4)( 5,6 , - 4 )(5,6,−4) Terminal point: (−4,2,7)( - 4,2,7 )(−4,2,7)

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Find the acute interior angle of the parallelogram formed by $A ( 2 , - 2,2 ) , B ( 5 , - 1,4 )$ , $C ( 6,0,10 )$ , and $D ( 3 , - 1,8 )$ . Round your answer to two decimals.

Find $\mathbf { u } \times \mathbf { v }$ . $\mathbf { u } = - 8 \mathbf { i } - 5 \mathbf { j } + 3 \mathbf { k } , \mathbf { v } = - 6 \mathbf { i } - 7 \mathbf { j } - 3 \mathbf { k }$

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 magnitude of the vector $\mathbf { v }$ . $\mathbf { v } = \langle 7 , - 4,0 \rangle$

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

Find the midpoint of the line segment joining the points. $( - 3,7 , - 8 ) , ( 8,2,5 )$

Find the acute interior angle of the parallelogram formed by $A ( 1 , - 1,2 ) , B ( 6,0,4 )$ , $C ( 7,1,9 )$ , and $D ( 2,0,7 )$ . Round your answer to two decimals.

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 described below. Initial point: $( 5,6 , - 4 )$ Terminal point: $( - 4,2,7 )$