Question 1

Find the angle between u and v in radians.
-u = -5i - 8j, v = 7i + 4j + 10k

Accepted Answer

A)  1.84 
B)  2.16 
C)  1.58 
D)  -0.59 
A)  1.84 
B)  2.16 
C)  1.58 
D)  -0.59

Question 2

Find the angle between u and v in radians.
-u = 3j - 5k, v = 7i - 6j - 3k

Accepted Answer

A)  1.62 
B)  -0.05 
C)  1.59 
D)  1.57 
A)  1.62 
B)  -0.05 
C)  1.59 
D)  1.57

Question 3

Match the equation with the surface it defines.
-$$\frac { z ^ { 2 } } { 9 } - \frac { x ^ { 2 } } { 81 } - \frac { y ^ { 2 } } { 81 } = 1$$

Accepted Answer

A)  Figure 2 
B)  Figure 1 
C)  Figure 4 
D)  Figure 3 
A)  Figure 2 
B)  Figure 1 
C)  Figure 4 
D)  Figure 3

Question 4

Solve the problem.
-An airplane is flying in the direction $9 ^ { \circ }$ east of south at $701 \mathrm {~km} / \mathrm { hr }$. Find the component form of the velocity of the airplane, assuming that the positive $x$-axis represents due east and the positive $y$-axis represents due north.

Accepted Answer

A)  $\langle 692.4 , - 109.7 \rangle$ 
B)  $\langle 109.7 , - 692.4 )$ 
C)  $\langle 0.1564 , - 0.987 \rangle \rangle$ 
D)  $\langle 288.9,638.7 \rangle$ 
A)  $\langle 692.4 , - 109.7 \rangle$ 
B)  $\langle 109.7 , - 692.4 )$ 
C)  $\langle 0.1564 , - 0.987 \rangle \rangle$ 
D)  $\langle 288.9,638.7 \rangle$

Question 5

Solve the problem.
-Find a formula for the distance from the point P(x, y, z) to the xz plane.

Accepted Answer

A)  $\sqrt { \mathrm { z } ^ { 2 } + \mathrm { x } ^ { 2 } }$ 
B)  $z$ 
C)  $y$ 
D)  $x$ 
A)  $\sqrt { \mathrm { z } ^ { 2 } + \mathrm { x } ^ { 2 } }$ 
B)  $z$ 
C)  $y$ 
D)  $x$

Question 6

Find the vector projv u.
-v = 3i - j + 3k, u = 10i + 11j + 2k

Accepted Answer

A)  $\frac { 141 } { 19 } \mathbf { i } - \frac { 47 } { 19 } \mathrm { j } + \frac { 141 } { 19 } \mathbf { k }$ 
B)  $\frac { 50 } { 3 } \mathrm { i } + \frac { 55 } { 3 } \mathrm { j } + \frac { 10 } { 3 } \mathrm { k }$ 
C)  $\frac { 75 } { 19 } \mathbf { i } - \frac { 25 } { 19 } \mathbf { j } + \frac { 75 } { 19 } \mathbf { k }$ 
D)  $\frac { 10 } { 9 } \mathrm { i } + \frac { 11 } { 9 } \mathrm { j } + \frac { 2 } { 9 } \mathbf { k }$ 
A)  $\frac { 141 } { 19 } \mathbf { i } - \frac { 47 } { 19 } \mathrm { j } + \frac { 141 } { 19 } \mathbf { k }$ 
B)  $\frac { 50 } { 3 } \mathrm { i } + \frac { 55 } { 3 } \mathrm { j } + \frac { 10 } { 3 } \mathrm { k }$ 
C)  $\frac { 75 } { 19 } \mathbf { i } - \frac { 25 } { 19 } \mathbf { j } + \frac { 75 } { 19 } \mathbf { k }$ 
D)  $\frac { 10 } { 9 } \mathrm { i } + \frac { 11 } { 9 } \mathrm { j } + \frac { 2 } { 9 } \mathbf { k }$

Question 7

Use a calculator to find the acute angle between the planes to the nearest thousandth of a radian.
-9x - 7y - 10z = -9 and- 3x + 10y - 10z = -9

Accepted Answer

A)  1.557 rad 
B)  1.308 rad 
C)  0.014 rad 
D)  1.535 rad 
A)  1.557 rad 
B)  1.308 rad 
C)  0.014 rad 
D)  1.535 rad

Question 8

Find the magnitude.
-Let $\mathbf { u } = \langle - 2,2 \rangle$ and $\mathbf { v } = \langle - 1,3 \rangle$. Find the magnitude (length) of the vector: $5 \mathbf { u } - \mathbf { v }$.

Accepted Answer

A)  123 
B)  130 
C)  $\sqrt { 123 }$ 
D)  $\sqrt { 130 }$ 
A)  123 
B)  130 
C)  $\sqrt { 123 }$ 
D)  $\sqrt { 130 }$

Question 9

Express the vector as a product of its length and direction.
-$$6 i - \frac { 4 } { 3 } j - 4 k$$

Accepted Answer

A)  $\frac { 22 } { 3 } ( \mathbf { i } - \mathbf { j } - \mathbf { k } )$ 
B)  $\frac { 22 } { 3 } \left( \frac { 9 } { 11 } \mathbf { i } - \frac { 2 } { 11 } \mathbf { j } - \frac { 6 } { 11 } \mathbf { k } \right)$ 
C)  $\frac { 22 } { 3 } \left( 6 \mathbf { i } - \frac { 4 } { 3 } \mathbf { j } - 4 \mathbf { k } \right)$ 
D)  $\frac { 22 } { 3 } \left( \frac { 9 } { 121 } \mathbf { i } - \frac { 2 } { 121 } \mathbf { j } - \frac { 6 } { 121 } \mathbf { k } \right)$ 
A)  $\frac { 22 } { 3 } ( \mathbf { i } - \mathbf { j } - \mathbf { k } )$ 
B)  $\frac { 22 } { 3 } \left( \frac { 9 } { 11 } \mathbf { i } - \frac { 2 } { 11 } \mathbf { j } - \frac { 6 } { 11 } \mathbf { k } \right)$ 
C)  $\frac { 22 } { 3 } \left( 6 \mathbf { i } - \frac { 4 } { 3 } \mathbf { j } - 4 \mathbf { k } \right)$ 
D)  $\frac { 22 } { 3 } \left( \frac { 9 } { 121 } \mathbf { i } - \frac { 2 } { 121 } \mathbf { j } - \frac { 6 } { 121 } \mathbf { k } \right)$

Question 10

Find the component form of the specified vector.
-The vector $\overrightarrow { \mathrm { PQ } }$, where $\mathrm { P } = ( - 10 , - 10 )$ and $\mathrm { Q } = ( - 2 , - 1 )$

Accepted Answer

A)  $\langle - 20 , - 1 \rangle$ 
B)  $( - 8 , - 9 \rangle$ 
C)  $\langle 8,9 \rangle$ 
D)  $\langle - 12 , - 11 \rangle$ 
A)  $\langle - 20 , - 1 \rangle$ 
B)  $( - 8 , - 9 \rangle$ 
C)  $\langle 8,9 \rangle$ 
D)  $\langle - 12 , - 11 \rangle$

Question 11

Write one or more inequalities that describe the set of points.
-The closed region bounded by the spheres of radius 2 and 7, both centered at the origin, and the planes x = 3 and x = 5

Accepted Answer

A)  $4 \leq x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 49$ and $3 \leq x \leq 5$ 
B)  $2 < x ^ { 2 } + y ^ { 2 } + z ^ { 2 } < 7$ and $x = 3$ and $x = 5$ 
C)  $2 \leq x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 7$ and $x = 3$ and $x = 5$ 
D)  $4 < x ^ { 2 } + y ^ { 2 } + z ^ { 2 } < 49$ and $3 < x < 5$ 
A)  $4 \leq x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 49$ and $3 \leq x \leq 5$ 
B)  $2 < x ^ { 2 } + y ^ { 2 } + z ^ { 2 } < 7$ and $x = 3$ and $x = 5$ 
C)  $2 \leq x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \leq 7$ and $x = 3$ and $x = 5$ 
D)  $4 < x ^ { 2 } + y ^ { 2 } + z ^ { 2 } < 49$ and $3 < x < 5$

Question 12

Describe the given set of points with a single equation or with a pair of equations.
-The circle in which the plane through the point (5, -8, -2) perpendicular to the x-axis meets the sphere of radius 13 centered at the origin.

Accepted Answer

A)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 169$ and $z = 5$ 
B)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 144$ and $y = 5$ 
C)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 119$ and $y = 5$ 
D)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 169$ and $x = 5$ 
A)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 169$ and $z = 5$ 
B)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 144$ and $y = 5$ 
C)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 119$ and $y = 5$ 
D)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 169$ and $x = 5$

Question 13

Solve the problem.
-For the vectors $\mathbf { u }$ and $\mathbf { v }$ with magnitudes $| \mathbf { u } | = 3$ and $| \mathbf { v } | = 6$, find the angle $\theta$ between $\mathbf { u }$ and $\mathbf { v }$ which makes $\mid$ proju $^ { \mathbf { v } } \mid = 2$

Accepted Answer

A)  $70.53 ^ { \circ }$ 
B)  $60.00 ^ { \circ }$ 
C)  $48.19 ^ { \circ }$ 
D)  $41.81 ^ { \circ }$ 
A)  $70.53 ^ { \circ }$ 
B)  $60.00 ^ { \circ }$ 
C)  $48.19 ^ { \circ }$ 
D)  $41.81 ^ { \circ }$

Question 14

Find the length and direction (when defined) of u × v.
-u = 6i, v = 7j

Accepted Answer

A)  6; 7k 
B)  42; 42k 
C)  42; k 
D)  42; -k 
A)  6; 7k 
B)  42; 42k 
C)  42; k 
D)  42; -k

Question 15

Determine whether the following is always true or not always true. Given reasons for your answers.
-u × 0 = 0

Accepted Answer

Always tru

Question 16

Find v · u.
-v = 8i - 2j and u = -3i + 6j

Accepted Answer

A)  5i + 4j 
B)  -36 
C)  -24i - 12j 
D)  -12 
A)  5i + 4j 
B)  -36 
C)  -24i - 12j 
D)  -12

Question 17

Identify the type of surface represented by the given equation.
-$$x = - 4 z ^ { 2 } , \text { no limit on } y$$

Accepted Answer

A)  Parabolic cylinder 
B)  Sphere 
C)  Hyperboloid of two sheets 
D)  Cylinder 
A)  Parabolic cylinder 
B)  Sphere 
C)  Hyperboloid of two sheets 
D)  Cylinder

Question 18

Find the magnitude.
-Let $\mathbf { u } = \langle - 1,2 \rangle$. Find the magnitude (length) of the vector: $7 \mathbf { u }$.

Accepted Answer

A)  $7 \sqrt { 10 }$ 
B)  $- 7 \sqrt { 5 }$ 
C)  $7 \sqrt { 5 }$ 
D)  $\sqrt { 10 }$ 
A)  $7 \sqrt { 10 }$ 
B)  $- 7 \sqrt { 5 }$ 
C)  $7 \sqrt { 5 }$ 
D)  $\sqrt { 10 }$

Question 19

$\text { Find the distance between points } P _ { 1 } \text { and } P _ { 2 } \text {. }$
-$P _ { 1 } ( 8,5 , - 1 )$ and $P _ { 2 } ( 9,6 , - 2 )$

Accepted Answer

A)  2 
B)  $\sqrt { 3 }$ 
C)  3 
D)  9 
A)  2 
B)  $\sqrt { 3 }$ 
C)  3 
D)  9

Question 20

Find the angle between u and v in radians.
-u = 10i - 5j - 9k, v = 4i + 7j - 5k

Accepted Answer

A)  1.40 
B)  1.19 
C)  1.57 
D)  0.38 
A)  1.40 
B)  1.19 
C)  1.57 
D)  0.38

Find the angle between u and v in radians. -u = -5i - 8j, v = 7i + 4j + 10k

Find the angle between u and v in radians. -u = 3j - 5k, v = 7i - 6j - 3k

Match the equation with the surface it defines. - $\frac { z ^ { 2 } } { 9 } - \frac { x ^ { 2 } } { 81 } - \frac { y ^ { 2 } } { 81 } = 1$ $Match the equation with the surface it defines. - \frac { z ^ { 2 } } { 9 } - \frac { x ^ { 2 } } { 81 } - \frac { y ^ { 2 } } { 81 } = 1$

Solve the problem. -An airplane is flying in the direction $9 ^ { \circ }$ east of south at $701 \mathrm {~km} / \mathrm { hr }$ . Find the component form of the velocity of the airplane, assuming that the positive $x$ -axis represents due east and the positive $y$ -axis represents due north.

Solve the problem. -Find a formula for the distance from the point P(x, y, z) to the xz plane.

Find the vector projv u. -v = 3i - j + 3k, u = 10i + 11j + 2k

Use a calculator to find the acute angle between the planes to the nearest thousandth of a radian. -9x - 7y - 10z = -9 and- 3x + 10y - 10z = -9

Find the magnitude. -Let $\mathbf { u } = \langle - 2,2 \rangle$ and $\mathbf { v } = \langle - 1,3 \rangle$ . Find the magnitude (length) of the vector: $5 \mathbf { u } - \mathbf { v }$ .

Express the vector as a product of its length and direction. - $6 i - \frac { 4 } { 3 } j - 4 k$

Find the component form of the specified vector. -The vector $\overrightarrow { \mathrm { PQ } }$ , where $\mathrm { P } = ( - 10 , - 10 )$ and $\mathrm { Q } = ( - 2 , - 1 )$

Write one or more inequalities that describe the set of points. -The closed region bounded by the spheres of radius 2 and 7, both centered at the origin, and the planes x = 3 and x = 5

Describe the given set of points with a single equation or with a pair of equations. -The circle in which the plane through the point (5, -8, -2) perpendicular to the x-axis meets the sphere of radius 13 centered at the origin.

Solve the problem. -For the vectors $\mathbf { u }$ and $\mathbf { v }$ with magnitudes $| \mathbf { u } | = 3$ and $| \mathbf { v } | = 6$ , find the angle $\theta$ between $\mathbf { u }$ and $\mathbf { v }$ which makes $\mid$ proju $^ { \mathbf { v } } \mid = 2$

Find the length and direction (when defined) of u × v. -u = 6i, v = 7j

Determine whether the following is always true or not always true. Given reasons for your answers. -u × 0 = 0

Find v · u. -v = 8i - 2j and u = -3i + 6j

Identify the type of surface represented by the given equation. - $x = - 4 z ^ { 2 } , \text { no limit on } y$

Find the magnitude. -Let $\mathbf { u } = \langle - 1,2 \rangle$ . Find the magnitude (length) of the vector: $7 \mathbf { u }$ .

$\text { Find the distance between points } P _ { 1 } \text { and } P _ { 2 } \text {. }$ - $P _ { 1 } ( 8,5 , - 1 )$ and $P _ { 2 } ( 9,6 , - 2 )$

Find the angle between u and v in radians. -u = 10i - 5j - 9k, v = 4i + 7j - 5k

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Exam 13: Vectors and the Geometry of Space