Question 1

Find the indicated vector.
-Let $\mathbf { u } = \langle - 7,3 \rangle$. Find $2 \mathbf { u }$.

Accepted Answer

A)  $\langle - 14 , - 6 \rangle$ 
B)  $\langle 14 , - 6 \rangle$ 
C)  $\langle 14,6 \rangle$ 
D)  $\langle - 14,6 \rangle$ 
A)  $\langle - 14 , - 6 \rangle$ 
B)  $\langle 14 , - 6 \rangle$ 
C)  $\langle 14,6 \rangle$ 
D)  $\langle - 14,6 \rangle$

Question 2

Identify the type of surface represented by the given equation.
-y2 + z2 = 7

Accepted Answer

A)  Paraboloid 
B)  Parabolic cylinder 
C)  Ellipsoid 
D)  Cylinder 
A)  Paraboloid 
B)  Parabolic cylinder 
C)  Ellipsoid 
D)  Cylinder

Question 3

Find the intersection.
-8x - 7y = -3, 4y + 7z = 5

Accepted Answer

A)  $x = - 49 t + \frac { 23 } { 32 } , y = - 56 t + \frac { 5 } { 4 } , z = 32$ 
B)  $x = - 49 t + 23 , y = - 56 t + 5 , z = 32 t$ 
C)  $x = - 49 t - \frac { 23 } { 32 } , y = - 56 t - \frac { 5 } { 4 } , z = - 32 t$ 
D)  $x = - 49 t + \frac { 23 } { 32 } , y = - 56 t + \frac { 5 } { 4 } , z = 32 t$ 
A)  $x = - 49 t + \frac { 23 } { 32 } , y = - 56 t + \frac { 5 } { 4 } , z = 32$ 
B)  $x = - 49 t + 23 , y = - 56 t + 5 , z = 32 t$ 
C)  $x = - 49 t - \frac { 23 } { 32 } , y = - 56 t - \frac { 5 } { 4 } , z = - 32 t$ 
D)  $x = - 49 t + \frac { 23 } { 32 } , y = - 56 t + \frac { 5 } { 4 } , z = 32 t$

Question 4

Find the indicated vector.
-Let $\mathbf { u } = \langle - 7 , - 4 \rangle$. Find $- 5 \mathbf { u }$.

Accepted Answer

A)  $\langle - 35 , - 20 \rangle$ 
B)  $\langle 35,20 \rangle$ 
C)  $\langle - 35,20 \rangle$ 
D)  $\langle 35 , - 20 \rangle$ 
A)  $\langle - 35 , - 20 \rangle$ 
B)  $\langle 35,20 \rangle$ 
C)  $\langle - 35,20 \rangle$ 
D)  $\langle 35 , - 20 \rangle$

Question 5

Solve the problem.
-Find the volume of the solid bounded by the paraboloid $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 81 } = \frac { z } { 2 }$ and the planes $z = 0$ and $z = 6$. (The area of an ellipse with semiaxes $\mathrm { a }$ and $\mathrm { b }$ is $\pi \mathrm { ab }$.)

Accepted Answer

A)  $567 \pi$ units $^ { 3 }$ 
B)  567 units $^ { 3 }$ 
C)  $1134 \pi$ units $^ { 3 }$ 
D)  $567 \pi ^ { 2 }$ units $^ { 3 }$ 
A)  $567 \pi$ units $^ { 3 }$ 
B)  567 units $^ { 3 }$ 
C)  $1134 \pi$ units $^ { 3 }$ 
D)  $567 \pi ^ { 2 }$ units $^ { 3 }$

Question 6

Express the vector as a product of its length and direction.
-$\frac { 8 } { 3 } \mathbf { j } - 2 \mathbf { k }$

Accepted Answer

A)  $\frac { 10 } { 3 } ( \mathrm { j } - \mathbf { k } )$ 
B)  $\frac { 100 } { 9 } \left( \frac { 4 } { 5 } \mathrm { j } - \frac { 3 } { 5 } \mathbf { k } \right)$ 
C)  $\frac { 10 } { 3 } \left( \frac { 4 } { 5 } \mathrm { j } - \frac { 3 } { 5 } \mathbf { k } \right)$ 
D)  $\frac { 10 } { 3 } \left( \frac { 8 } { 3 } \mathbf { j } - 2 \mathbf { k } \right)$ 
A)  $\frac { 10 } { 3 } ( \mathrm { j } - \mathbf { k } )$ 
B)  $\frac { 100 } { 9 } \left( \frac { 4 } { 5 } \mathrm { j } - \frac { 3 } { 5 } \mathbf { k } \right)$ 
C)  $\frac { 10 } { 3 } \left( \frac { 4 } { 5 } \mathrm { j } - \frac { 3 } { 5 } \mathbf { k } \right)$ 
D)  $\frac { 10 } { 3 } \left( \frac { 8 } { 3 } \mathbf { j } - 2 \mathbf { k } \right)$

Question 7

Find the center and radius of the sphere.
-$$( x + 2 ) ^ { 2 } + ( y + 8 ) ^ { 2 } + ( z + 4 ) ^ { 2 } = \frac { 1 } { 16 }$$

Accepted Answer

A)  $\mathrm { C } ( - 2 , - 8 , - 4 ) , \mathrm { a } = \frac { 1 } { 4 }$ 
B)  $\mathrm { C } ( 2,8,4 ) , \mathrm { a } = \frac { 1 } { 16 }$ 
C)  $C ( - 2 , - 8 , - 4 ) , a = \frac { 1 } { 16 }$ 
D)  $\mathrm { C } ( 2,8,4 ) , \mathrm { a } = \frac { 1 } { 4 }$ 
A)  $\mathrm { C } ( - 2 , - 8 , - 4 ) , \mathrm { a } = \frac { 1 } { 4 }$ 
B)  $\mathrm { C } ( 2,8,4 ) , \mathrm { a } = \frac { 1 } { 16 }$ 
C)  $C ( - 2 , - 8 , - 4 ) , a = \frac { 1 } { 16 }$ 
D)  $\mathrm { C } ( 2,8,4 ) , \mathrm { a } = \frac { 1 } { 4 }$

Question 8

Find the vector projv u.
-v = 7i - 3j + k, u = -4j + 3k

Accepted Answer

A)  $\frac { 105 } { 59 } \mathbf { i } - \frac { 45 } { 59 } \mathbf { j } + \frac { 15 } { 59 } \mathbf { k }$ 
B)  $- \frac { 60 } { 59 } \mathbf { j } + \frac { 45 } { 59 } \mathbf { k }$ 
C)  $\frac { 21 } { 5 } \mathbf { i } - \frac { 9 } { 5 } \mathbf { j } + \frac { 3 } { 5 } \mathbf { k }$ 
D)  $- \frac { 12 } { 5 } \mathbf { j } + \frac { 9 } { 5 } \mathbf { k }$ 
A)  $\frac { 105 } { 59 } \mathbf { i } - \frac { 45 } { 59 } \mathbf { j } + \frac { 15 } { 59 } \mathbf { k }$ 
B)  $- \frac { 60 } { 59 } \mathbf { j } + \frac { 45 } { 59 } \mathbf { k }$ 
C)  $\frac { 21 } { 5 } \mathbf { i } - \frac { 9 } { 5 } \mathbf { j } + \frac { 3 } { 5 } \mathbf { k }$ 
D)  $- \frac { 12 } { 5 } \mathbf { j } + \frac { 9 } { 5 } \mathbf { k }$

Question 9

Find the intersection.
-x = -2 + 2t, y = 1 + 10t, z = -2 + 7t ; -10x + 2y + 8z = 4

Accepted Answer

A)  $\left( - \frac { 29 } { 14 } , \frac { 9 } { 14 } , - \frac { 9 } { 4 } \right)$ 
B)  $( 0,11,5 )$ 
C)  $\left( - \frac { 27 } { 14 } , \frac { 19 } { 14 } , \frac { 5 } { 4 } \right)$ 
D)  $( - 4 , - 9 , - 9 )$ 
A)  $\left( - \frac { 29 } { 14 } , \frac { 9 } { 14 } , - \frac { 9 } { 4 } \right)$ 
B)  $( 0,11,5 )$ 
C)  $\left( - \frac { 27 } { 14 } , \frac { 19 } { 14 } , \frac { 5 } { 4 } \right)$ 
D)  $( - 4 , - 9 , - 9 )$

Question 10

Find an equation for the line that passes through the given point and satisfies the given conditions.
-P = (-8, 3); perpendicular to v = -2i - 5j

Accepted Answer

A)  $- 5 x + 2 y = 46$ 
B)  $- 2 x - 5 y = 29$ 
C)  $y - 3 = - \frac { 4 } { 3 } ( x + 2 )$ 
D)  $- 2 x - 5 y = 1$ 
A)  $- 5 x + 2 y = 46$ 
B)  $- 2 x - 5 y = 29$ 
C)  $y - 3 = - \frac { 4 } { 3 } ( x + 2 )$ 
D)  $- 2 x - 5 y = 1$

Question 11

Solve the problem.
-A bullet is fired with a muzzle velocity of $1324 \mathrm { ft } / \mathrm { sec }$ from a gun aimed at an angle of $36 ^ { \circ }$ above the horizontal. Find the horizontal component of the velocity.

Accepted Answer

A)  1071 ft/sec 
B)  961.9 ft/sec 
C)  778.2 ft/sec 
D)  -169.4 ft/sec 
A)  1071 ft/sec 
B)  961.9 ft/sec 
C)  778.2 ft/sec 
D)  -169.4 ft/sec

Question 12

Solve the problem.
-$\text { Find the magnitude of the torque in foot-pounds at point } P \text { for the following lever: }$
 
$|\overrightarrow{P Q}|=6 \text { in. and }|F|=10 \mathrm{lb}$

Accepted Answer

A)  6.76 ft-lb 
B)  4.53 ft-lb 
C)  2.11 ft-lb 
D)  60 ft-lb 
A)  6.76 ft-lb 
B)  4.53 ft-lb 
C)  2.11 ft-lb 
D)  60 ft-lb

Question 13

Solve the problem.
-Find the volume of the solid bounded by the hyperboloid of one sheet $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 64 } - \frac { z ^ { 2 } } { 9 } = 1$ and the planes $z = - 2$ and $z = 2$. (The area of an ellipse with semiaxes a and $b$ is $\pi a b$.)

Accepted Answer

A)  $103.11 \pi$ units $^ { 3 }$ 
B)  $110.22 \pi$ units $^ { 3 }$ 
C)  $81.78 \pi$ units $^ { 3 }$ 
D)  $2645.33 \pi$ units $^ { 3 }$ 
A)  $103.11 \pi$ units $^ { 3 }$ 
B)  $110.22 \pi$ units $^ { 3 }$ 
C)  $81.78 \pi$ units $^ { 3 }$ 
D)  $2645.33 \pi$ units $^ { 3 }$

Question 14

Use a calculator to find the acute angle between the planes to the nearest thousandth of a radian.
-8x + 4y + 3z = 1 and 7x + 10y + 7z = -6

Accepted Answer

A)  1.079 rad 
B)  0.492 rad 
C)  1.534 rad 
D)  1.458 rad 
A)  1.079 rad 
B)  0.492 rad 
C)  1.534 rad 
D)  1.458 rad

Question 15

Find the center and radius of the sphere.
-$$x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 8 x - 2 y + 8 z = 3$$

Accepted Answer

A)  C(4, 1, 4), a = 6 
B)  C(4, 1, -4), a = 6 
C)  C(4, 1, -4), a = 36 
D)  C(-4, -1, 4), a = 6 
A)  C(4, 1, 4), a = 6 
B)  C(4, 1, -4), a = 6 
C)  C(4, 1, -4), a = 36 
D)  C(-4, -1, 4), a = 6

Question 16

Find the indicated vector.
-Let $\mathbf { u } = \langle - 2 , - 5 \rangle , \mathbf { v } = \langle - 2,5 \rangle$. Find $\mathbf { v } - \mathbf { u }$.

Accepted Answer

A)  $\langle - 4,0 \rangle$ 
B)  $\langle - 3,7 \rangle$ 
C)  $\langle 0,10 \rangle$ 
D)  $\langle 7,3 \rangle$ 
A)  $\langle - 4,0 \rangle$ 
B)  $\langle - 3,7 \rangle$ 
C)  $\langle 0,10 \rangle$ 
D)  $\langle 7,3 \rangle$

Question 17

Calculate the requested distance.
-The distance from the point S(-10, -4, -6) to the plane -9x + 2y + 6z

Accepted Answer

A)  $\frac { 122 } { 11 }$ 
B)  $\frac { 42 } { 11 }$ 
C)  $\frac { 122 } { 121 }$ 
D)  $\frac { 42 } { 121 }$ 
A)  $\frac { 122 } { 11 }$ 
B)  $\frac { 42 } { 11 }$ 
C)  $\frac { 122 } { 121 }$ 
D)  $\frac { 42 } { 121 }$

Question 18

$$\text { Use the vectors } u , v , w \text {, and } z \text { head to tail as needed to sketch the indicated vector. }$$

-$\mathbf{Z}=\mathbf{V}$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 19

Solve the problem.
-For the triangle with vertices located at A(3, 5, 5), B(5, 2, 4), and C(1, 1, 1) , find a vector from vertex C to the midpoint of side AB.

Accepted Answer

A)  $4 \mathbf { i } + \frac { 7 } { 2 } \mathbf { j } + \frac { 9 } { 2 } \mathbf { k }$ 
B)  $5 \mathbf { i } + \frac { 9 } { 2 } \mathrm { j } + \frac { 11 } { 2 } \mathbf { k }$ 
C)  $\frac { 1 } { 2 } \mathrm { i } + \frac { 3 } { 2 } \mathrm { j } + \frac { 3 } { 2 } \mathrm { k }$ 
D)  $3 i + \frac { 5 } { 2 } j + \frac { 7 } { 2 } k$ 
A)  $4 \mathbf { i } + \frac { 7 } { 2 } \mathbf { j } + \frac { 9 } { 2 } \mathbf { k }$ 
B)  $5 \mathbf { i } + \frac { 9 } { 2 } \mathrm { j } + \frac { 11 } { 2 } \mathbf { k }$ 
C)  $\frac { 1 } { 2 } \mathrm { i } + \frac { 3 } { 2 } \mathrm { j } + \frac { 3 } { 2 } \mathrm { k }$ 
D)  $3 i + \frac { 5 } { 2 } j + \frac { 7 } { 2 } k$

Question 20

Find the triple scalar product (u x v) · w of the given vectors.
-u = 2i - 4j + 3k; v = -4i - 7j + 9k; w = 7i - 3j + 3k

Accepted Answer

A)  495 
B)  -3 
C)  142 
D)  -105 
A)  495 
B)  -3 
C)  142 
D)  -105

Find the indicated vector. -Let $\mathbf { u } = \langle - 7,3 \rangle$ . Find $2 \mathbf { u }$ .

Identify the type of surface represented by the given equation. -y2 + z2 = 7

Find the intersection. -8x - 7y = -3, 4y + 7z = 5

Find the indicated vector. -Let $\mathbf { u } = \langle - 7 , - 4 \rangle$ . Find $- 5 \mathbf { u }$ .

Solve the problem. -Find the volume of the solid bounded by the paraboloid $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 81 } = \frac { z } { 2 }$ and the planes $z = 0$ and $z = 6$ . (The area of an ellipse with semiaxes $\mathrm { a }$ and $\mathrm { b }$ is $\pi \mathrm { ab }$ .)

Express the vector as a product of its length and direction. - $\frac { 8 } { 3 } \mathbf { j } - 2 \mathbf { k }$

Find the center and radius of the sphere. - $( x + 2 ) ^ { 2 } + ( y + 8 ) ^ { 2 } + ( z + 4 ) ^ { 2 } = \frac { 1 } { 16 }$

Find the vector projv u. -v = 7i - 3j + k, u = -4j + 3k

Find the intersection. -x = -2 + 2t, y = 1 + 10t, z = -2 + 7t ; -10x + 2y + 8z = 4

Find an equation for the line that passes through the given point and satisfies the given conditions. -P = (-8, 3); perpendicular to v = -2i - 5j

Solve the problem. -A bullet is fired with a muzzle velocity of $1324 \mathrm { ft } / \mathrm { sec }$ from a gun aimed at an angle of $36 ^ { \circ }$ above the horizontal. Find the horizontal component of the velocity.

Solve the problem. -Find the volume of the solid bounded by the hyperboloid of one sheet $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 64 } - \frac { z ^ { 2 } } { 9 } = 1$ and the planes $z = - 2$ and $z = 2$ . (The area of an ellipse with semiaxes a and $b$ is $\pi a b$ .)

Use a calculator to find the acute angle between the planes to the nearest thousandth of a radian. -8x + 4y + 3z = 1 and 7x + 10y + 7z = -6

Find the center and radius of the sphere. - $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 8 x - 2 y + 8 z = 3$

Find the indicated vector. -Let $\mathbf { u } = \langle - 2 , - 5 \rangle , \mathbf { v } = \langle - 2,5 \rangle$ . Find $\mathbf { v } - \mathbf { u }$ .

Calculate the requested distance. -The distance from the point S(-10, -4, -6) to the plane -9x + 2y + 6z

Solve the problem. -For the triangle with vertices located at A(3, 5, 5), B(5, 2, 4), and C(1, 1, 1) , find a vector from vertex C to the midpoint of side AB.

Find the triple scalar product (u x v) · w of the given vectors. -u = 2i - 4j + 3k; v = -4i - 7j + 9k; w = 7i - 3j + 3k

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Exam 13: Vectors and the Geometry of Space

Find the indicated vector. -Let u=⟨−7,3⟩\mathbf { u } = \langle - 7,3 \rangleu=⟨−7,3⟩ . Find 2u2 \mathbf { u }2u .