Question 1

The polar coordinates of a point are given. Find the rectangular coordinates of the point.
-$\left( - 9 , \frac { 2 \pi } { 3 } \right)$

Accepted Answer

A)  $\left( \frac { 9 } { 2 } , \frac { 9 \sqrt { 3 } } { 2 } \right)$ 
B)  $\left( - \frac { 9 } { 2 } , \frac { 9 \sqrt { 3 } } { 2 } \right)$ 
C)  $\left( \frac { 9 } { 2 } , \frac { - 9 \sqrt { 3 } } { 2 } \right)$ 
D)  $\left( - \frac { 9 } { 2 } , \frac { - 9 \sqrt { 3 } } { 2 } \right)$ 
A)  $\left( \frac { 9 } { 2 } , \frac { 9 \sqrt { 3 } } { 2 } \right)$ 
B)  $\left( - \frac { 9 } { 2 } , \frac { 9 \sqrt { 3 } } { 2 } \right)$ 
C)  $\left( \frac { 9 } { 2 } , \frac { - 9 \sqrt { 3 } } { 2 } \right)$ 
D)  $\left( - \frac { 9 } { 2 } , \frac { - 9 \sqrt { 3 } } { 2 } \right)$ C

Question 2

The letters x and y represent rectangular coordinates. Write the equation using polar coordinates (r, θ).
-$$x ^ { 2 } + y ^ { 2 } - 4 x = 0$$

Accepted Answer

A)  $r = 4 \cos \theta$ 
B)  $r \cos ^ { 2 } \theta = 4 \sin \theta$ 
C)  $r \sin ^ { 2 } \theta = 4 \cos \theta$ 
D)  $r = 4 \sin \theta$ 
A)  $r = 4 \cos \theta$ 
B)  $r \cos ^ { 2 } \theta = 4 \sin \theta$ 
C)  $r \sin ^ { 2 } \theta = 4 \cos \theta$ 
D)  $r = 4 \sin \theta$ A

Question 3

Find all the complex roots. Leave your answers in polar form with the argument in degrees.
-The complex cube roots of $- 8 \mathrm { i }$

Accepted Answer

A)  $2 \left( \cos 90 ^ { \circ } + i \sin 90 ^ { \circ } \right) , 2 \left( \cos 210 ^ { \circ } + i \sin 210 ^ { \circ } \right) , 2 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$ 
B)  $512 \left( \cos 90 ^ { \circ } + i \sin 90 ^ { \circ } \right) , 512 \left( \cos 210 ^ { \circ } + i \sin 210 ^ { \circ } \right) , 512 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$ 
C)  $8 \left( \cos 90 ^ { \circ } + i \sin 90 ^ { \circ } \right) , 8 \left( \cos 210 ^ { \circ } + i \sin 210 ^ { \circ } \right) , 8 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$ 
D)  $2 \left( \cos 180 ^ { \circ } + i \sin 180 ^ { \circ } \right) , 2 \left( \cos 300 ^ { \circ } + i \sin 300 ^ { \circ } \right) , 2 \left( \cos 60 ^ { \circ } + i \sin 60 ^ { \circ } \right)$ 
A)  $2 \left( \cos 90 ^ { \circ } + i \sin 90 ^ { \circ } \right) , 2 \left( \cos 210 ^ { \circ } + i \sin 210 ^ { \circ } \right) , 2 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$ 
B)  $512 \left( \cos 90 ^ { \circ } + i \sin 90 ^ { \circ } \right) , 512 \left( \cos 210 ^ { \circ } + i \sin 210 ^ { \circ } \right) , 512 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$ 
C)  $8 \left( \cos 90 ^ { \circ } + i \sin 90 ^ { \circ } \right) , 8 \left( \cos 210 ^ { \circ } + i \sin 210 ^ { \circ } \right) , 8 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$ 
D)  $2 \left( \cos 180 ^ { \circ } + i \sin 180 ^ { \circ } \right) , 2 \left( \cos 300 ^ { \circ } + i \sin 300 ^ { \circ } \right) , 2 \left( \cos 60 ^ { \circ } + i \sin 60 ^ { \circ } \right)$ A

Question 4

Perform the indicated operation.
-v = 6i + 4j + 3k and w = 5i + 2j - 6k Find 4v - 3w.

Accepted Answer

A)  -9i - 2j + 21k 
B)  9i + 10j + 30k 
C)  29i + 18j + 6k 
D)  11i + 6j - 3k 
A)  -9i - 2j + 21k 
B)  9i + 10j + 30k 
C)  29i + 18j + 6k 
D)  11i + 6j - 3k

Question 5

Find the requested vector.
-v = 4i + 2j + k, u = -2i + 2j - 2k Find a vector orthogonal to both v and u.

Accepted Answer

A)  -5i + 9j + 2k 
B)  5i - 9j + 9k 
C)  4i - 12j - 16k 
D)  -6i + 6j + 12k 
A)  -5i + 9j + 2k 
B)  5i - 9j + 9k 
C)  4i - 12j - 16k 
D)  -6i + 6j + 12k

Question 6

Plot the point given in polar coordinates.
-$$\left( - 2,45 ^ { \circ } \right)$$
 
 A)

Accepted Answer

B)

C) 
  
D) 
  
B)

C) 
  
D)

Question 7

Match the graph to one of the polar equations.
-  A) $r = 2 \sin \theta$

Accepted Answer

B)  $r = 2 \cos \theta$ 
C)  $r \sin \theta = 1$ 
D)  $r = 1$ 
B)  $r = 2 \cos \theta$ 
C)  $r \sin \theta = 1$ 
D)  $r = 1$

Question 8

Find the unit vector having the same direction as v.
-$$v = - 30 i + 30 j - 15 k$$

Accepted Answer

A)  $\frac { v } { \| v } = - \frac { 2 } { \sqrt { 5 } } \mathrm { i } + \frac { 2 } { \sqrt { 5 } } \mathrm { j } - \frac { 1 } { \sqrt { 5 } } \mathbf { k }$ 
B)  $\frac { v } { \| v \| } = \frac { 2 } { 3 } \mathbf { i } + \frac { 2 } { 3 } j + \frac { 1 } { 3 } k$ 
C)  $\frac { \mathbf { v } } { \| \mathbf { v } \| } = - \frac { 2 } { 9 } \mathbf { i } + \frac { 2 } { 9 } \mathbf { j } - \frac { 1 } { 9 } \mathbf { k }$ 
D)  $\frac { v } { v \| } = - \frac { 2 } { 3 } \mathbf { i } + \frac { 2 } { 3 } \mathbf { j } - \frac { 1 } { 3 } \mathbf { k }$ 
A)  $\frac { v } { \| v } = - \frac { 2 } { \sqrt { 5 } } \mathrm { i } + \frac { 2 } { \sqrt { 5 } } \mathrm { j } - \frac { 1 } { \sqrt { 5 } } \mathbf { k }$ 
B)  $\frac { v } { \| v \| } = \frac { 2 } { 3 } \mathbf { i } + \frac { 2 } { 3 } j + \frac { 1 } { 3 } k$ 
C)  $\frac { \mathbf { v } } { \| \mathbf { v } \| } = - \frac { 2 } { 9 } \mathbf { i } + \frac { 2 } { 9 } \mathbf { j } - \frac { 1 } { 9 } \mathbf { k }$ 
D)  $\frac { v } { v \| } = - \frac { 2 } { 3 } \mathbf { i } + \frac { 2 } { 3 } \mathbf { j } - \frac { 1 } { 3 } \mathbf { k }$

Question 9

Find the indicated cross product.
-v = 5i - 6j, w = 3i + 2j - 2k Find w × v.

Accepted Answer

A)  -4i + 6j + 36k 
B)  -12i - 10j - 28k 
C)  -28i + 12j - 12k 
D)  10i + 12j - 28k 
A)  -4i + 6j + 36k 
B)  -12i - 10j - 28k 
C)  -28i + 12j - 12k 
D)  10i + 12j - 28k

Question 10

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

Accepted Answer

A)  $x ^ { 2 } + y ^ { 2 } = ( 1 + 2 x ) ^ { 2 }$ 
B)  $x ^ { 2 } + y ^ { 2 } = 1 + 2 x$ 
C)  $x ^ { 2 } + y ^ { 2 } = 2 + x$ 
D)  $x ^ { 2 } + y ^ { 2 } = ( 2 + x ) ^ { 2 }$ 
A)  $x ^ { 2 } + y ^ { 2 } = ( 1 + 2 x ) ^ { 2 }$ 
B)  $x ^ { 2 } + y ^ { 2 } = 1 + 2 x$ 
C)  $x ^ { 2 } + y ^ { 2 } = 2 + x$ 
D)  $x ^ { 2 } + y ^ { 2 } = ( 2 + x ) ^ { 2 }$

Question 11

Find the dot product v · w.
-v = 2i + j + 3k and w = i + 2j - 2k

Accepted Answer

A)  -2 
B)  -7 
C)  2 
D)  -6 
A)  -2 
B)  -7 
C)  2 
D)  -6

Question 12

Test the equation for symmetry with respect to the given axis, line, or pole.
-$r = 2 \sin \theta$; 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 13

The rectangular coordinates of a point are given. Find polar coordinates for the point.
-(0.6, -1.1) Round the polar coordinates to two decimal places, with θ in degrees.

Accepted Answer

A)  (1.25, -61.39°) 
B)  (1.25, 57.93°) 
C)  (1.25, -57.93°) 
D)  (1.25, 61.39°) 
A)  (1.25, -61.39°) 
B)  (1.25, 57.93°) 
C)  (1.25, -57.93°) 
D)  (1.25, 61.39°)

Question 14

Plot the point given in polar coordinates.
-$\left[5, \frac{5 \pi}{3}\right)$

A)

Accepted Answer

B)

C) 
  
D) 
  
B)

C) 
  
D)

Question 15

Find the direction angles of the vector. Round to the nearest degree, if necessary.
-$$\mathbf { v } = 4 \mathbf { i } + 2 \mathrm { j } - 3 \mathbf { k }$$

Accepted Answer

A)  $\alpha = 14 ^ { \circ } , \beta = 61 ^ { \circ } , \gamma = 137 ^ { \circ }$ 
B)  $\alpha = 42 ^ { \circ } , \beta = 68 ^ { \circ } , \gamma = 124 ^ { \circ }$ 
C)  $\alpha = 40 ^ { \circ } , \beta = 67 ^ { \circ } , \gamma = 125 ^ { \circ }$ 
D)  $\alpha = 33 ^ { \circ } , \beta = 65 ^ { \circ } , \gamma = 129 ^ { \circ }$ 
A)  $\alpha = 14 ^ { \circ } , \beta = 61 ^ { \circ } , \gamma = 137 ^ { \circ }$ 
B)  $\alpha = 42 ^ { \circ } , \beta = 68 ^ { \circ } , \gamma = 124 ^ { \circ }$ 
C)  $\alpha = 40 ^ { \circ } , \beta = 67 ^ { \circ } , \gamma = 125 ^ { \circ }$ 
D)  $\alpha = 33 ^ { \circ } , \beta = 65 ^ { \circ } , \gamma = 129 ^ { \circ }$

Question 16

Perform the indicated operation.
-v = 4i - 5j + 6k and w = 3i - 4j + 2k Find 3v + 2w.

Accepted Answer

A)  10i - 13j + 10k 
B)  7i - 9j + 8k 
C)  15i - 19j + 20k 
D)  18i - 23j + 22k 
A)  10i - 13j + 10k 
B)  7i - 9j + 8k 
C)  15i - 19j + 20k 
D)  18i - 23j + 22k

Question 17

Test the equation for symmetry with respect to the given axis, line, or pole.
-$r = 4 \cos \theta ;$ the line $\theta = \frac { \pi } { 2 }$

Accepted Answer

A)  May or may not be symmetric with respect to the line $\theta = \frac { \pi } { 2 }$ 
B)  Symmetric with respect to the line $\theta = \frac { \pi } { 2 }$ 
A)  May or may not be symmetric with respect to the line $\theta = \frac { \pi } { 2 }$ 
B)  Symmetric with respect to the line $\theta = \frac { \pi } { 2 }$

Question 18

Find the area of the parallelogram.
-$P _ { 1 } ( 0,0,0 ) , P _ { 2 } ( 2 , - 4,1 ) , P _ { 3 } ( - 1,2 , - 1 )$

Accepted Answer

A)  $\sqrt { 6 }$ 
B)  $\sqrt { 41 }$ 
C)  $\sqrt { 5 }$ 
D)  $\sqrt { 21 }$ 
A)  $\sqrt { 6 }$ 
B)  $\sqrt { 41 }$ 
C)  $\sqrt { 5 }$ 
D)  $\sqrt { 21 }$

Question 19

Plot the complex number in the complex plane.
-$-4-6 i$
 
 A)

Accepted Answer

B)

C) 
  
D) 
  
B)

C) 
  
D)

Question 20

Graph the polar equation.
-$r=2-2 \cos \theta$
 
 A)

Accepted Answer

B)

C) 
  
D) 
  
B)

C) 
  
D)

The polar coordinates of a point are given. Find the rectangular coordinates of the point. - $\left( - 9 , \frac { 2 \pi } { 3 } \right)$

The letters x and y represent rectangular coordinates. Write the equation using polar coordinates (r, θ). - $x ^ { 2 } + y ^ { 2 } - 4 x = 0$

Find all the complex roots. Leave your answers in polar form with the argument in degrees. -The complex cube roots of $- 8 \mathrm { i }$

Perform the indicated operation. -v = 6i + 4j + 3k and w = 5i + 2j - 6k Find 4v - 3w.

Find the requested vector. -v = 4i + 2j + k, u = -2i + 2j - 2k Find a vector orthogonal to both v and u.

Match the graph to one of the polar equations. - $Match the graph to one of the polar equations. - A) r = 2 \sin \theta B) r = 2 \cos \theta C) r \sin \theta = 1 D) r = 1$ A) $r = 2 \sin \theta$ B) $r = 2 \cos \theta$ C) $r \sin \theta = 1$ D) $r = 1$

Find the unit vector having the same direction as v. - $v = - 30 i + 30 j - 15 k$

Find the indicated cross product. -v = 5i - 6j, w = 3i + 2j - 2k Find w × v.

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

Find the dot product v · w. -v = 2i + j + 3k and w = i + 2j - 2k

Test the equation for symmetry with respect to the given axis, line, or pole. - $r = 2 \sin \theta$ ; the pole

The rectangular coordinates of a point are given. Find polar coordinates for the point. -(0.6, -1.1) Round the polar coordinates to two decimal places, with θ in degrees.

Find the direction angles of the vector. Round to the nearest degree, if necessary. - $\mathbf { v } = 4 \mathbf { i } + 2 \mathrm { j } - 3 \mathbf { k }$

Perform the indicated operation. -v = 4i - 5j + 6k and w = 3i - 4j + 2k Find 3v + 2w.

Test the equation for symmetry with respect to the given axis, line, or pole. - $r = 4 \cos \theta ;$ the line $\theta = \frac { \pi } { 2 }$

Find the area of the parallelogram. - $P _ { 1 } ( 0,0,0 ) , P _ { 2 } ( 2 , - 4,1 ) , P _ { 3 } ( - 1,2 , - 1 )$

Plot the complex number in the complex plane. - $-4-6 i$ A) B) C) D)

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

Review

Filters

Exam 8: Polar Coordinates; Vectors

The polar coordinates of a point are given. Find the rectangular coordinates of the point. - (−9,2π3)\left( - 9 , \frac { 2 \pi } { 3 } \right)(−9,32π​)

The letters x and y represent rectangular coordinates. Write the equation using polar coordinates (r, θ). - x2+y2−4x=0x ^ { 2 } + y ^ { 2 } - 4 x = 0x2+y2−4x=0

Find all the complex roots. Leave your answers in polar form with the argument in degrees. -The complex cube roots of −8i- 8 \mathrm { i }−8i

Perform the indicated operation. -v = 6i + 4j + 3k and w = 5i + 2j - 6k Find 4v - 3w.

Find the requested vector. -v = 4i + 2j + k, u = -2i + 2j - 2k Find a vector orthogonal to both v and u.

Plot the point given in polar coordinates. - (−2,45∘)\left( - 2,45 ^ { \circ } \right)(−2,45∘) A) B) C) D)

Match the graph to one of the polar equations. - A) r=2sin⁡θr = 2 \sin \thetar=2sinθ B) r=2cos⁡θr = 2 \cos \thetar=2cosθ C) rsin⁡θ=1r \sin \theta = 1rsinθ=1 D) r=1r = 1r=1

Find the unit vector having the same direction as v. - v=−30i+30j−15kv = - 30 i + 30 j - 15 kv=−30i+30j−15k

Find the indicated cross product. -v = 5i - 6j, w = 3i + 2j - 2k Find w × v.

The letters r and θ represent polar coordinates. Write the equation using rectangular coordinates (x, y). - r(1−2cos⁡θ)=1r ( 1 - 2 \cos \theta ) = 1r(1−2cosθ)=1

Find the dot product v · w. -v = 2i + j + 3k and w = i + 2j - 2k

Test the equation for symmetry with respect to the given axis, line, or pole. - r=2sin⁡θr = 2 \sin \thetar=2sinθ ; the pole

The rectangular coordinates of a point are given. Find polar coordinates for the point. -(0.6, -1.1) Round the polar coordinates to two decimal places, with θ in degrees.

Plot the point given in polar coordinates. - [5,5π3)\left[5, \frac{5 \pi}{3}\right)[5,35π​) A) B) C) D)

Find the direction angles of the vector. Round to the nearest degree, if necessary. - v=4i+2j−3k\mathbf { v } = 4 \mathbf { i } + 2 \mathrm { j } - 3 \mathbf { k }v=4i+2j−3k

Perform the indicated operation. -v = 4i - 5j + 6k and w = 3i - 4j + 2k Find 3v + 2w.

Test the equation for symmetry with respect to the given axis, line, or pole. - r=4cos⁡θ;r = 4 \cos \theta ;r=4cosθ; the line θ=π2\theta = \frac { \pi } { 2 }θ=2π​

Find the area of the parallelogram. - P1(0,0,0),P2(2,−4,1),P3(−1,2,−1)P _ { 1 } ( 0,0,0 ) , P _ { 2 } ( 2 , - 4,1 ) , P _ { 3 } ( - 1,2 , - 1 )P1​(0,0,0),P2​(2,−4,1),P3​(−1,2,−1)

Plot the complex number in the complex plane. - −4−6i-4-6 i−4−6i A) B) C) D)

Graph the polar equation. - r=2−2cos⁡θr=2-2 \cos \thetar=2−2cosθ A) B) C) D)

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

Review

Filters

The polar coordinates of a point are given. Find the rectangular coordinates of the point. - $\left( - 9 , \frac { 2 \pi } { 3 } \right)$

The letters x and y represent rectangular coordinates. Write the equation using polar coordinates (r, θ). - $x ^ { 2 } + y ^ { 2 } - 4 x = 0$

Find all the complex roots. Leave your answers in polar form with the argument in degrees. -The complex cube roots of $- 8 \mathrm { i }$

Match the graph to one of the polar equations. - $Match the graph to one of the polar equations. - A) r = 2 \sin \theta B) r = 2 \cos \theta C) r \sin \theta = 1 D) r = 1$ A) $r = 2 \sin \theta$ B) $r = 2 \cos \theta$ C) $r \sin \theta = 1$ D) $r = 1$

Find the unit vector having the same direction as v. - $v = - 30 i + 30 j - 15 k$

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

Test the equation for symmetry with respect to the given axis, line, or pole. - $r = 2 \sin \theta$ ; the pole

Find the direction angles of the vector. Round to the nearest degree, if necessary. - $\mathbf { v } = 4 \mathbf { i } + 2 \mathrm { j } - 3 \mathbf { k }$

Test the equation for symmetry with respect to the given axis, line, or pole. - $r = 4 \cos \theta ;$ the line $\theta = \frac { \pi } { 2 }$

Find the area of the parallelogram. - $P _ { 1 } ( 0,0,0 ) , P _ { 2 } ( 2 , - 4,1 ) , P _ { 3 } ( - 1,2 , - 1 )$

Plot the complex number in the complex plane. - $-4-6 i$ A) B) C) D)