Question 1

Use the vectors in the figure below to graph the following vector.  
-$v - w$

Accepted Answer

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

Question 2

Identify and graph the polar equation.
-$r=1-\cos \theta$
  A)
 
cardioid

Accepted Answer

B) 
 
cardioid
 
C) 
 
cardioid 
D) 
 
cardioid 
B) 
 
cardioid
 
C) 
 
cardioid 
D) 
 
cardioid

Question 3

Find $$\| \mathbf { v } \|$$ .
-$$\mathbf { v } = 6 \mathbf { i } + 5 \mathbf { j } - 6 \mathbf { k }$$

Accepted Answer

A)  $\sqrt { 97 }$ 
B)  5 
C)  $\sqrt { 61 }$ 
D)  $\sqrt { 77 }$ 
A)  $\sqrt { 97 }$ 
B)  5 
C)  $\sqrt { 61 }$ 
D)  $\sqrt { 77 }$

Question 4

Write the complex number in polar form. Express the argument in degrees, rounded to the nearest tenth, if necessary.
-$- 4 + 4 \mathrm { i }$

Accepted Answer

A)  $4 \sqrt { 2 } \left( \cos 120 ^ { \circ } + i \sin 120 ^ { \circ } \right)$ 
B)  $4 \sqrt { 2 } \left( \cos 135 ^ { \circ } + i \sin 135 ^ { \circ } \right)$ 
C)  $16 \left( \cos 135 ^ { \circ } + i \sin 135 ^ { \circ } \right)$ 
D)  $16 \left( \cos 120 ^ { \circ } + \mathrm { i } \sin 120 ^ { \circ } \right)$ 
A)  $4 \sqrt { 2 } \left( \cos 120 ^ { \circ } + i \sin 120 ^ { \circ } \right)$ 
B)  $4 \sqrt { 2 } \left( \cos 135 ^ { \circ } + i \sin 135 ^ { \circ } \right)$ 
C)  $16 \left( \cos 135 ^ { \circ } + i \sin 135 ^ { \circ } \right)$ 
D)  $16 \left( \cos 120 ^ { \circ } + \mathrm { i } \sin 120 ^ { \circ } \right)$

Question 5

Write the vector v in the form ai + bj, given its magnitude v and the angle α it makes with the positive x-axis.
-$$\| \mathbf { v } \| = 15 , \alpha = 210 ^ { \circ }$$

Accepted Answer

A)  $\mathbf { v } = 15 \left( \frac { 15 \sqrt { 3 } } { 2 } \mathrm { i } - \frac { 15 } { 2 } \mathrm { j } \right)$ 
B)  $v = 15 \left( - \frac { \sqrt { 2 } } { 2 } \mathrm { i } - \frac { \sqrt { 2 } } { 2 } \mathrm { j } \right)$ 
C)  $\mathbf { v } = 15 \left( - \frac { 15 \sqrt { 3 } } { 2 } \mathrm { i } - \frac { 1 } { 2 } \mathrm { j } \right)$ 
D)  $\mathbf { v } = 15 \left( - \frac { 15 } { 2 } \mathbf { i } - \frac { 15 \sqrt { 3 } } { 2 } \mathrm { j } \right)$ 
A)  $\mathbf { v } = 15 \left( \frac { 15 \sqrt { 3 } } { 2 } \mathrm { i } - \frac { 15 } { 2 } \mathrm { j } \right)$ 
B)  $v = 15 \left( - \frac { \sqrt { 2 } } { 2 } \mathrm { i } - \frac { \sqrt { 2 } } { 2 } \mathrm { j } \right)$ 
C)  $\mathbf { v } = 15 \left( - \frac { 15 \sqrt { 3 } } { 2 } \mathrm { i } - \frac { 1 } { 2 } \mathrm { j } \right)$ 
D)  $\mathbf { v } = 15 \left( - \frac { 15 } { 2 } \mathbf { i } - \frac { 15 \sqrt { 3 } } { 2 } \mathrm { j } \right)$

Question 6

The letters x and y represent rectangular coordinates. Write the equation using polar coordinates (r, θ).
-$$2 x + 3 y = 6$$

Accepted Answer

A)  $2 \sin \theta + 3 \cos \theta = 6 r$ 
B)  $2 \cos \theta + 3 \sin \theta = 6 r$ 
C)  $\mathrm { r } ( 2 \cos \theta + 3 \sin \theta ) = 6$ 
D)  $r ( 2 \sin \theta + 3 \cos \theta ) = 6$ 
A)  $2 \sin \theta + 3 \cos \theta = 6 r$ 
B)  $2 \cos \theta + 3 \sin \theta = 6 r$ 
C)  $\mathrm { r } ( 2 \cos \theta + 3 \sin \theta ) = 6$ 
D)  $r ( 2 \sin \theta + 3 \cos \theta ) = 6$

Question 7

Plot the complex number in the complex plane.
-$6+\sqrt{7} i$
 
 A)

Accepted Answer

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

Question 8

Plot the point given in polar coordinates.
-$$\left( 4,315 ^ { \circ } \right)$$
 
 A)

Accepted Answer

B)

C) 
  
D) 
  
B)

C) 
  
D)

Question 9

Solve the problem.
-If u = -9i - 2j and v = -3i + 7j, find u - v.

Accepted Answer

A)  -6i - 9j 
B)  -7i + 5j 
C)  -8i + 5j 
D)  -12i + 5j 
A)  -6i - 9j 
B)  -7i + 5j 
C)  -8i + 5j 
D)  -12i + 5j

Question 10

The rectangular coordinates of a point are given. Find polar coordinates for the point.
-$( - 7,0 )$

Accepted Answer

A)  $\left( 7 , \frac { \pi } { 2 } \right)$ 
B)  $\left( 7 , \frac { 3 \pi } { 2 } \right)$ 
C)  $( 7 , \pi )$ 
D)  $( - 7 , \pi )$ 
A)  $\left( 7 , \frac { \pi } { 2 } \right)$ 
B)  $\left( 7 , \frac { 3 \pi } { 2 } \right)$ 
C)  $( 7 , \pi )$ 
D)  $( - 7 , \pi )$

Question 11

The letters r and θ represent polar coordinates. Write the equation using rectangular coordinates (x, y).
-$$\mathrm { r } = 1 + 2 \sin \theta$$

Accepted Answer

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

Question 12

State whether the vectors are parallel, orthogonal, or neither.
-v = 4i - 2j, w = 4i + 2j

Accepted Answer

A)  Orthogonal 
B)  Parallel 
C)  Neither 
A)  Orthogonal 
B)  Parallel 
C)  Neither

Question 13

Solve the problem.
-Two forces of magnitude 25 pounds and 40 pounds act on an object. The force of 40 lb acts along the positive x-axis, and the force of 25 lb acts at an angle of 80° with the positive x-axis. Find the direction and magnitude
Of the resultant force. Round the direction and magnitude to the nearest whole number.

Accepted Answer

A)  Direction: 40°; magnitude: 47 lb 
B)  Direction: 51°; magnitude: 51 lb 
C)  Direction: 29°; magnitude: 51 lb 
D)  Direction: 4°; magnitude: 65 lb 
A)  Direction: 40°; magnitude: 47 lb 
B)  Direction: 51°; magnitude: 51 lb 
C)  Direction: 29°; magnitude: 51 lb 
D)  Direction: 4°; magnitude: 65 lb

Question 14

The polar coordinates of a point are given. Find the rectangular coordinates of the point.
-(400, 130°) Round the rectangular coordinates to two decimal places.

Accepted Answer

A)  (306.42, -257.12) 
B)  (-257.12, -306.42) 
C)  (-257.12, 306.42) 
D)  (306.42, 257.12) 
A)  (306.42, -257.12) 
B)  (-257.12, -306.42) 
C)  (-257.12, 306.42) 
D)  (306.42, 257.12)

Question 15

Plot the point given in polar coordinates.
-$$\left( 3 , \frac { 7 \pi } { 6 } \right)$$
 
 A)

Accepted Answer

B)

C) 
  
D) 
  
B)

C) 
  
D)

Question 16

Solve the problem. Leave your answer in polar form.
-$$\begin{array} { l } 
z = 8 \left( \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \right) \
W = 3 \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)
\end{array}$$
Find zw.

Accepted Answer

A)  $24 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$ 
B)  $24 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$ 
C)  $12 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$ 
D)  $12 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$ 
A)  $24 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$ 
B)  $24 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$ 
C)  $12 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$ 
D)  $12 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$

Question 17

Find the direction angles of the vector. Round to the nearest degree, if necessary.
-$$\mathbf { v } = - 6 \mathbf { i } - 4 \mathbf { j } - 12 \mathbf { k }$$

Accepted Answer

A)  $\alpha = 117 ^ { \circ } , \beta = 108 ^ { \circ } , \gamma = 157 ^ { \circ }$ 
B)  $\alpha = 115 ^ { \circ } , \beta = 107 ^ { \circ } , \gamma = 149 ^ { \circ }$ 
C)  $\alpha = 118 ^ { \circ } , \beta = 108 ^ { \circ } , \gamma = 161 ^ { \circ }$ 
D)  $\alpha = 92 ^ { \circ } , \beta = 91 ^ { \circ } , \gamma = 94 ^ { \circ }$ 
A)  $\alpha = 117 ^ { \circ } , \beta = 108 ^ { \circ } , \gamma = 157 ^ { \circ }$ 
B)  $\alpha = 115 ^ { \circ } , \beta = 107 ^ { \circ } , \gamma = 149 ^ { \circ }$ 
C)  $\alpha = 118 ^ { \circ } , \beta = 108 ^ { \circ } , \gamma = 161 ^ { \circ }$ 
D)  $\alpha = 92 ^ { \circ } , \beta = 91 ^ { \circ } , \gamma = 94 ^ { \circ }$

Question 18

Use the given vectors to find the indicated expression.
-v = -4i - 3j + 2k, w = -5i - 5j + 4k, u = 4i + 4j - 4k Find w · (v × u).

Accepted Answer

A)  148 
B)  4 
C)  172 
D)  0 
A)  148 
B)  4 
C)  172 
D)  0

Question 19

Test the equation for symmetry with respect to the given axis, line, or pole.
-$$\mathrm { r } ^ { 2 } = \sin ( 2 \theta ) \text {; the pole }$$

Accepted Answer

A)  Symmetric with respect to the pole 
B)  May or may not be symmetric with respect to the pole 
A)  Symmetric with respect to the pole 
B)  May or may not be symmetric with respect to the pole

Question 20

Transform the polar equation to an equation in rectangular coordinates. Then identify and graph the equation.
-$$r = 6 \sin \theta$$
 
 A)
 
$ x^{2}+(y-3)^{2}=9 ; $ circle, radius 3 center at $ (0,3) $ in rectangular coordinates

Accepted Answer

B) 
 
$$
(x+3)^{2}+y^{2}=9 \text {; circle, radius 3 }
$$
center at $ (-3,0) $ in rectangular coordinates 
C) 
 
$ (x-3)^{2}+y^{2}=9 $; circle, radius 3 , center at $ (3,0) $ in rectangular coordinates 
D) 
 
$ \mathrm{x}^{2}+(\mathrm{y}+3)^{2}=9 $; circle, radius 3 center at $ (0,-3) $ in rectangular coordinates 
B) 
 
$$
(x+3)^{2}+y^{2}=9 \text {; circle, radius 3 }
$$
center at $ (-3,0) $ in rectangular coordinates 
C) 
 
$ (x-3)^{2}+y^{2}=9 $; circle, radius 3 , center at $ (3,0) $ in rectangular coordinates 
D) 
 
$ \mathrm{x}^{2}+(\mathrm{y}+3)^{2}=9 $; circle, radius 3 center at $ (0,-3) $ in rectangular coordinates

Use the vectors in the figure below to graph the following vector. - $v - w$

Find $\| \mathbf { v } \|$ . - $\mathbf { v } = 6 \mathbf { i } + 5 \mathbf { j } - 6 \mathbf { k }$

Write the complex number in polar form. Express the argument in degrees, rounded to the nearest tenth, if necessary. - $- 4 + 4 \mathrm { i }$

Write the vector v in the form ai + bj, given its magnitude v and the angle α it makes with the positive x-axis. - $\| \mathbf { v } \| = 15 , \alpha = 210 ^ { \circ }$

The letters x and y represent rectangular coordinates. Write the equation using polar coordinates (r, θ). - $2 x + 3 y = 6$

Solve the problem. -If u = -9i - 2j and v = -3i + 7j, find u - v.

The rectangular coordinates of a point are given. Find polar coordinates for the point. - $( - 7,0 )$

The letters r and θ represent polar coordinates. Write the equation using rectangular coordinates (x, y). - $\mathrm { r } = 1 + 2 \sin \theta$

State whether the vectors are parallel, orthogonal, or neither. -v = 4i - 2j, w = 4i + 2j

The polar coordinates of a point are given. Find the rectangular coordinates of the point. -(400, 130°) Round the rectangular coordinates to two decimal places.

Solve the problem. Leave your answer in polar form. - z=8 +i W=3 +i Find zw.

Find the direction angles of the vector. Round to the nearest degree, if necessary. - $\mathbf { v } = - 6 \mathbf { i } - 4 \mathbf { j } - 12 \mathbf { k }$

Use the given vectors to find the indicated expression. -v = -4i - 3j + 2k, w = -5i - 5j + 4k, u = 4i + 4j - 4k Find w · (v × u).

Test the equation for symmetry with respect to the given axis, line, or pole. - $\mathrm { r } ^ { 2 } = \sin ( 2 \theta ) \text {; the pole }$

Functions and Their Graphs

Linear and Quadratic Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometric Functions

Analytic Geometry

Systems of Equations and Inequalities

Sequences; Induction; the Binomial Theorem

Counting and Probability

A Preview of Calculus: the Limit, Derivative, and Integral of a Function

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Exam 8: Polar Coordinates; Vectors

Use the vectors in the figure below to graph the following vector. - v−wv - wv−w

Identify and graph the polar equation. - r=1−cos⁡θr=1-\cos \thetar=1−cosθ A) cardioid B) cardioid C) cardioid D) cardioid

Find ∥v∥\| \mathbf { v } \|∥v∥ . - v=6i+5j−6k\mathbf { v } = 6 \mathbf { i } + 5 \mathbf { j } - 6 \mathbf { k }v=6i+5j−6k

Write the complex number in polar form. Express the argument in degrees, rounded to the nearest tenth, if necessary. - −4+4i- 4 + 4 \mathrm { i }−4+4i

Write the vector v in the form ai + bj, given its magnitude v and the angle α it makes with the positive x-axis. - ∥v∥=15,α=210∘\| \mathbf { v } \| = 15 , \alpha = 210 ^ { \circ }∥v∥=15,α=210∘

The letters x and y represent rectangular coordinates. Write the equation using polar coordinates (r, θ). - 2x+3y=62 x + 3 y = 62x+3y=6

Plot the complex number in the complex plane. - 6+7i6+\sqrt{7} i6+7​i A) B) C) D)

Plot the point given in polar coordinates. - (4,315∘)\left( 4,315 ^ { \circ } \right)(4,315∘) A) B) C) D)