Question 1

Find the vector projv u.
-v = i + j + k, u = 3i + 4j + 12k

Accepted Answer

A)  $\frac { 19 } { 169 } \mathbf { i } + \frac { 19 } { 169 } \mathbf { j } + \frac { 19 } { 169 } \mathbf { k }$ 
B)  $\frac { 19 } { 13 } \mathbf { i } + \frac { 19 } { 13 } \mathbf { j } + \frac { 19 } { 13 } \mathbf { k }$ 
C)  $\frac { 20 } { 3 } \mathbf { i } + \frac { 20 } { 3 } \mathbf { j } + \frac { 20 } { 3 } \mathbf { k }$ 
D)  $\frac { 19 } { 3 } \mathbf { i } + \frac { 19 } { 3 } \mathbf { j } + \frac { 19 } { 3 } \mathbf { k }$ 
A)  $\frac { 19 } { 169 } \mathbf { i } + \frac { 19 } { 169 } \mathbf { j } + \frac { 19 } { 169 } \mathbf { k }$ 
B)  $\frac { 19 } { 13 } \mathbf { i } + \frac { 19 } { 13 } \mathbf { j } + \frac { 19 } { 13 } \mathbf { k }$ 
C)  $\frac { 20 } { 3 } \mathbf { i } + \frac { 20 } { 3 } \mathbf { j } + \frac { 20 } { 3 } \mathbf { k }$ 
D)  $\frac { 19 } { 3 } \mathbf { i } + \frac { 19 } { 3 } \mathbf { j } + \frac { 19 } { 3 } \mathbf { k }$

Question 2

Identify the type of surface represented by the given equation.
-$$\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 3 } = 8$$

Accepted Answer

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

Question 3

Find the component form of the specified vector.
-The vector from the point A(8, 1) to the origin

Accepted Answer

A)  $\langle 8,1 \rangle$ 
B)  $\langle - 8 , - 1 \rangle$ 
C)  $\langle - 8,1 \rangle$ 
D)  $\langle 8 , - 1 \rangle$ 
A)  $\langle 8,1 \rangle$ 
B)  $\langle - 8 , - 1 \rangle$ 
C)  $\langle - 8,1 \rangle$ 
D)  $\langle 8 , - 1 \rangle$

Question 4

Find an equation for the line that passes through the given point and satisfies the given conditions.
-P = (-5, 7); parallel to v = -2i - 3j

Accepted Answer

A)  $- 2 x - 3 y = - 11$ 
B)  $- 2 x - 3 y = 13$ 
C)  $y - 7 = - \frac { 10 } { 3 } ( x + 2 )$ 
D)  $- 3 x + 2 y = 29$ 
A)  $- 2 x - 3 y = - 11$ 
B)  $- 2 x - 3 y = 13$ 
C)  $y - 7 = - \frac { 10 } { 3 } ( x + 2 )$ 
D)  $- 3 x + 2 y = 29$

Question 5

Identify the type of surface represented by the given equation.
-$$\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 3 } - \frac { z ^ { 2 } } { 4 } = 1$$

Accepted Answer

A)  Hyperboloid of two sheets 
B)  Hyperboloid of one sheet 
C)  Ellipsoid 
D)  Elliptical cone 
A)  Hyperboloid of two sheets 
B)  Hyperboloid of one sheet 
C)  Ellipsoid 
D)  Elliptical cone

Question 6

Find the indicated vector.
-Let $\mathbf { u } = \langle - 7 , - 5 \rangle , \mathbf { v } = \langle 7 , - 3 \rangle$. Find $\frac { 5 } { 13 } \mathbf { u } - \frac { 12 } { 13 } \mathbf { v }$.

Accepted Answer

A)  $\left( \frac { 25 } { 13 } , \frac { 71 } { 13 } \right)$ 
B)  $\left( \frac { 11 } { 13 } , - \frac { 119 } { 13 } \right)$ 
C)  $\left( - \frac { 119 } { 13 } , \frac { 11 } { 13 } \right)$ 
D)  $\left( - \frac { 70 } { 13 } , - \frac { 24 } { 31 } \right)$ 
A)  $\left( \frac { 25 } { 13 } , \frac { 71 } { 13 } \right)$ 
B)  $\left( \frac { 11 } { 13 } , - \frac { 119 } { 13 } \right)$ 
C)  $\left( - \frac { 119 } { 13 } , \frac { 11 } { 13 } \right)$ 
D)  $\left( - \frac { 70 } { 13 } , - \frac { 24 } { 31 } \right)$

Question 7

Solve the problem.
-A force of magnitude 10 pounds pulling on a suitcase makes an angle of $60 ^ { \circ }$ with the ground. Express the force in terms of its $\mathbf { i }$ and $\mathbf { j }$ components.

Accepted Answer

A)  0.5000i + + 0.8660j 
B)  5.000i + 8.660 j 
C)  -9.524i - 3.048j 
D)  8.660i + 5.000j 
A)  0.5000i + + 0.8660j 
B)  5.000i + 8.660 j 
C)  -9.524i - 3.048j 
D)  8.660i + 5.000j

Question 8

Find an equation for the sphere with the given center and radius.
-Center (-2, 0, 0), radius = 9

Accepted Answer

A)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x = 77$ 
B)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 4 x = 77$ 
C)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x = 9$ 
D)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 4 x = 9$ 
A)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x = 77$ 
B)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 4 x = 77$ 
C)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x = 9$ 
D)  $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 4 x = 9$

Question 9

Express the vector as a product of its length and direction.
-$$\frac { 1 } { \sqrt { 6 } } \mathbf { i } + \frac { 1 } { \sqrt { 3 } } \mathbf { j } - \frac { 1 } { \sqrt { 2 } } \mathbf { k }$$

Accepted Answer

A)  $\frac { 1 } { 6 } \left( \frac { 6 } { \sqrt { 6 } } \mathbf { i } + \frac { 6 } { \sqrt { 3 } } \mathbf { j } - \frac { 6 } { \sqrt { 2 } } \mathbf { k } \right)$ 
B)  $1 \left( \frac { 1 } { \sqrt { 6 } } \mathbf { i } + \frac { 1 } { \sqrt { 3 } } \mathbf { j } - \frac { 1 } { \sqrt { 2 } } \mathbf { k } \right)$ 
C)  $\frac { 1 } { 2 } \left( \frac { 1 } { 3 } \mathbf { i } + \frac { 2 } { \sqrt { 3 } } \mathbf { j } - \frac { 2 } { \sqrt { 2 } } \mathbf { k } \right)$ 
D)  $\frac { 2 } { 3 } \left( \frac { 3 } { 2 \sqrt { 6 } } \mathrm { i } + \frac { 3 } { 2 \sqrt { 3 } } \mathrm { j } - \frac { 3 } { 2 \sqrt { 2 } } \mathbf { k } \right)$ 
A)  $\frac { 1 } { 6 } \left( \frac { 6 } { \sqrt { 6 } } \mathbf { i } + \frac { 6 } { \sqrt { 3 } } \mathbf { j } - \frac { 6 } { \sqrt { 2 } } \mathbf { k } \right)$ 
B)  $1 \left( \frac { 1 } { \sqrt { 6 } } \mathbf { i } + \frac { 1 } { \sqrt { 3 } } \mathbf { j } - \frac { 1 } { \sqrt { 2 } } \mathbf { k } \right)$ 
C)  $\frac { 1 } { 2 } \left( \frac { 1 } { 3 } \mathbf { i } + \frac { 2 } { \sqrt { 3 } } \mathbf { j } - \frac { 2 } { \sqrt { 2 } } \mathbf { k } \right)$ 
D)  $\frac { 2 } { 3 } \left( \frac { 3 } { 2 \sqrt { 6 } } \mathrm { i } + \frac { 3 } { 2 \sqrt { 3 } } \mathrm { j } - \frac { 3 } { 2 \sqrt { 2 } } \mathbf { k } \right)$

Question 10

Describe the given set of points with a single equation or with a pair of equations.
-The plane through the point (-1, -2, -5) and perpendicular to the x-axis

Accepted Answer

A)  y = -2 and z = -5 
B)  -1 + y + z = 0 
C)  x = -1 
D)  y + z = -7 
A)  y = -2 and z = -5 
B)  -1 + y + z = 0 
C)  x = -1 
D)  y + z = -7

Question 11

Match the equation with the surface it defines.
-$$\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 81 } = \frac { z ^ { 2 } } { 100 }$$

Accepted Answer

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

Question 12

Find parametric equations for the line described below.
-The line through the point P( , , 0) and perpendicular to the plane x + 6y + 4z = 3

Accepted Answer

A)  x = -3t - 3, y = -6t - 6, z = -4t 
B)  x = -6t + 3, y = -6t + 3, z = 0 
C)  x = 3t - 3, y = 6t - 6, z = 4t 
D)  x = 3t + 3, y = 6t + 6, z = 4t 
A)  x = -3t - 3, y = -6t - 6, z = -4t 
B)  x = -6t + 3, y = -6t + 3, z = 0 
C)  x = 3t - 3, y = 6t - 6, z = 4t 
D)  x = 3t + 3, y = 6t + 6, z = 4t

Question 13

Match the equation with the surface it defines.
-$$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 100 } + \frac { z ^ { 2 } } { 25 } = 1$$

Accepted Answer

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

Question 14

Solve the problem.
-How much work does it take to slide a box 46 meters along the ground by pulling it with a $72 \mathrm {~N}$ force at an angle of $15 ^ { \circ }$ from the horizontal?

Accepted Answer

A)  3312 joules 
B)  69.55 joules 
C)  3199 joules 
D)  -2516 joules 
A)  3312 joules 
B)  69.55 joules 
C)  3199 joules 
D)  -2516 joules

Question 15

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

Accepted Answer

A)  $\frac { 4 } { 9 } \mathbf { i } + \frac { 1 } { 3 } \mathbf { j } + \frac { 1 } { 3 } \mathbf { k }$ 
B)  $\frac { 3 } { 10 } \mathrm { j }$ 
C)  $\frac { 4 } { 9 } \mathrm { i } + \frac { 1 } { 3 } \mathrm { j }$ 
D)  0 
A)  $\frac { 4 } { 9 } \mathbf { i } + \frac { 1 } { 3 } \mathbf { j } + \frac { 1 } { 3 } \mathbf { k }$ 
B)  $\frac { 3 } { 10 } \mathrm { j }$ 
C)  $\frac { 4 } { 9 } \mathrm { i } + \frac { 1 } { 3 } \mathrm { j }$ 
D)  0

Question 16

Solve the problem.
-Find the work done by a force of 18i (newtons) in moving as object along a line from the origin to the point (3, 3) (distance in meters).

Accepted Answer

A)  54 joules 
B)  $54 \sqrt { 2 }$ joules 
C)  $38.18$ joules 
D)  $18 \sqrt { 2 }$ joules 
A)  54 joules 
B)  $54 \sqrt { 2 }$ joules 
C)  $38.18$ joules 
D)  $18 \sqrt { 2 }$ joules

Question 17

$\text { Express the vector in the form } v = v _ { 1 } i + v _ { 2 } j + v _ { 3 } k \text {. }$
-$\overrightarrow { \mathrm { P } _ { 1 } \mathrm { P } _ { 2 } } \text { if } \mathrm { P } _ { 1 } \text { is the point } ( - 6 , - 3,4 ) \text { and } \mathrm { P } _ { 2 } \text { is the point } ( - 4 , - 6,0 )$

Accepted Answer

A)  v = -2i + 3j - 4k 
B)  v = -2i - 3j + 4k 
C)  v = -2i + 3j + 4k 
D)  v = 2i - 3j - 4k 
A)  v = -2i + 3j - 4k 
B)  v = -2i - 3j + 4k 
C)  v = -2i + 3j + 4k 
D)  v = 2i - 3j - 4k

Question 18

Find the acute angle, in degrees, between the lines.
-3x - y = 16 and 2x + y = -13

Accepted Answer

A)  $30 ^ { \circ }$ 
B)  $75 ^ { \circ }$ 
C)  $45 ^ { \circ }$ 
D)  $60 ^ { \circ }$ 
A)  $30 ^ { \circ }$ 
B)  $75 ^ { \circ }$ 
C)  $45 ^ { \circ }$ 
D)  $60 ^ { \circ }$

Question 19

Solve the problem.
-Let $\mathbf { u } = - 2 \mathbf { i } + 7 \mathbf { j } , \mathbf { v } = 2 \mathbf { i } + 3 \mathbf { j }$, and $\mathbf { w } = \mathbf { i } - \mathbf { j }$. Write $\mathbf { u } = \mathbf { u } _ { 1 } + \mathbf { u } _ { 2 }$ where $\mathbf { u } _ { 1 }$ is parallel to $\mathbf { v }$ and $\mathbf { u } _ { 2 }$ is parallel to $\mathbf { w }$.

Accepted Answer

A)  $\mathbf { u } _ { 1 } = 3 \mathrm { v }$
$u _ { 2 } = 2 w$ 
B)  $\mathbf { u } _ { 1 } = - 4 \mathrm { v }$
$u _ { 2 } = - 21 w$ 
C)  $\mathbf { u } _ { 1 } = 1.000 \mathbf { v }$
$\mathbf { u } _ { 2 } = - 0.2500 \mathbf { w }$ 
D)  $\mathrm { u } _ { 1 } = 1.000 \mathrm { v }$
$u _ { 2 } = - 4.000 \mathrm { w }$ 
A)  $\mathbf { u } _ { 1 } = 3 \mathrm { v }$
$u _ { 2 } = 2 w$ 
B)  $\mathbf { u } _ { 1 } = - 4 \mathrm { v }$
$u _ { 2 } = - 21 w$ 
C)  $\mathbf { u } _ { 1 } = 1.000 \mathbf { v }$
$\mathbf { u } _ { 2 } = - 0.2500 \mathbf { w }$ 
D)  $\mathrm { u } _ { 1 } = 1.000 \mathrm { v }$
$u _ { 2 } = - 4.000 \mathrm { w }$

Question 20

Give a geometric description of the set of points whose coordinates satisfy the given conditions.
-$$( x - 10 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + ( z - 2 ) ^ { 2 } < 4$$

Accepted Answer

A)  No set of points satisfy the given relations. 
B)  All points on the lower hemisphere centered at (10, 2, 2) 
C)  All points within the lower hemisphere centered at (10, 2, 2) 
D)  All points outside the lower hemisphere centered at (10, 2, 2) 
A)  No set of points satisfy the given relations. 
B)  All points on the lower hemisphere centered at (10, 2, 2) 
C)  All points within the lower hemisphere centered at (10, 2, 2) 
D)  All points outside the lower hemisphere centered at (10, 2, 2)

Find the vector projv u. -v = i + j + k, u = 3i + 4j + 12k

Identify the type of surface represented by the given equation. - $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 3 } = 8$

Find the component form of the specified vector. -The vector from the point A(8, 1) to the origin

Find an equation for the line that passes through the given point and satisfies the given conditions. -P = (-5, 7); parallel to v = -2i - 3j

Identify the type of surface represented by the given equation. - $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 3 } - \frac { z ^ { 2 } } { 4 } = 1$

Find the indicated vector. -Let $\mathbf { u } = \langle - 7 , - 5 \rangle , \mathbf { v } = \langle 7 , - 3 \rangle$ . Find $\frac { 5 } { 13 } \mathbf { u } - \frac { 12 } { 13 } \mathbf { v }$ .

Solve the problem. -A force of magnitude 10 pounds pulling on a suitcase makes an angle of $60 ^ { \circ }$ with the ground. Express the force in terms of its $\mathbf { i }$ and $\mathbf { j }$ components.

Find an equation for the sphere with the given center and radius. -Center (-2, 0, 0), radius = 9

Express the vector as a product of its length and direction. - $\frac { 1 } { \sqrt { 6 } } \mathbf { i } + \frac { 1 } { \sqrt { 3 } } \mathbf { j } - \frac { 1 } { \sqrt { 2 } } \mathbf { k }$

Describe the given set of points with a single equation or with a pair of equations. -The plane through the point (-1, -2, -5) and perpendicular to the x-axis

Match the equation with the surface it defines. - $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 81 } = \frac { z ^ { 2 } } { 100 }$ $Match the equation with the surface it defines. - \frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 81 } = \frac { z ^ { 2 } } { 100 }$

Find parametric equations for the line described below. -The line through the point P( , , 0) and perpendicular to the plane x + 6y + 4z = 3

Match the equation with the surface it defines. - $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 100 } + \frac { z ^ { 2 } } { 25 } = 1$ $Match the equation with the surface it defines. - \frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 100 } + \frac { z ^ { 2 } } { 25 } = 1$

Solve the problem. -How much work does it take to slide a box 46 meters along the ground by pulling it with a $72 \mathrm {~N}$ force at an angle of $15 ^ { \circ }$ from the horizontal?

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

Solve the problem. -Find the work done by a force of 18i (newtons) in moving as object along a line from the origin to the point (3, 3) (distance in meters).

Find the acute angle, in degrees, between the lines. -3x - y = 16 and 2x + y = -13

Give a geometric description of the set of points whose coordinates satisfy the given conditions. - $( x - 10 ) ^ { 2 } + ( y - 2 ) ^ { 2 } + ( z - 2 ) ^ { 2 } < 4$

Functions

Limits and Continuity

Derivatives

Applications of Derivatives

Integrals

Applications of Definite Integrals

Integrals and Transcendental Functions

Techniques of Integration

First-Order Differential Equations

Infinite Sequences and Series

Parametric Equations and Polar Coordinates

Vector-Valued Functions and Motion in Space

Partial Derivatives

Multiple Integrals

Integrals and Vector Fields

Filters

Exam 13: Vectors and the Geometry of Space