Question 1

The rectangular coordinates of a point are given. Find polar coordinates for the point.
-$( 0,9 )$

Accepted Answer

A)  $( 9,0 )$ 
B)  $\left( 9 , - \frac { \pi } { 2 } \right)$ 
C)  $( 9 , \pi )$ 
D)  $\left( 9 , \frac { \pi } { 2 } \right)$ 
A)  $( 9,0 )$ 
B)  $\left( 9 , - \frac { \pi } { 2 } \right)$ 
C)  $( 9 , \pi )$ 
D)  $\left( 9 , \frac { \pi } { 2 } \right)$

Question 2

Identify and graph the polar equation.
-$$r = 5 - 4 \cos \theta$$

Accepted Answer

A) 
  
B) 
 
 limacon without inner loop
limacon without inner loop 
C) 
  
D) 
 
 limacon with inner loop
limacon with inner loop 
A) 
  
B) 
 
 limacon without inner loop
limacon without inner loop 
C) 
  
D) 
 
 limacon with inner loop
limacon with inner loop

Question 3

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

Accepted Answer

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

Question 4

The vector v has initial position P and terminal point Q. Write v in the form ai + bj; that is, find its position vector.
-$$\mathrm { P } = ( 0,0 ) ; \mathrm { Q } = ( 6,4 )$$

Accepted Answer

A)  $\mathbf { v } = - 6 \mathbf { i } - 4 \mathbf { j }$ 
B)  $\mathbf { v } = - 4 \mathbf { i } - 6 \mathbf { j }$ 
C)  $\mathbf { v } = 4 \mathbf { i } + 4 \mathbf { j }$ 
D)  $\mathbf { v } = 6 i + 4 j$ 
A)  $\mathbf { v } = - 6 \mathbf { i } - 4 \mathbf { j }$ 
B)  $\mathbf { v } = - 4 \mathbf { i } - 6 \mathbf { j }$ 
C)  $\mathbf { v } = 4 \mathbf { i } + 4 \mathbf { j }$ 
D)  $\mathbf { v } = 6 i + 4 j$

Question 5

Choose the one alternative that best completes the statement or answers the question.
-Which of the following vectors is parallel to $\mathbf { v } = 9 \mathbf { i } + 5 \mathbf { j }$ ?

Accepted Answer

A)  $w = - \frac { 15 } { 2 } \mathrm { i } + \frac { 25 } { 6 } \mathrm { j }$ 
B)  $w = 45 i - 25 j$ 
C)  $\mathbf { w } = \frac { 15 } { 2 } \mathbf { i } - \frac { 25 } { 6 } \mathbf { j }$ 
D)  $w = \frac { 3 } { 2 } i + \frac { 5 } { 6 } j$ 
A)  $w = - \frac { 15 } { 2 } \mathrm { i } + \frac { 25 } { 6 } \mathrm { j }$ 
B)  $w = 45 i - 25 j$ 
C)  $\mathbf { w } = \frac { 15 } { 2 } \mathbf { i } - \frac { 25 } { 6 } \mathbf { j }$ 
D)  $w = \frac { 3 } { 2 } i + \frac { 5 } { 6 } j$

Question 6

Write the complex number in rectangular form.
-$9 \left( \cos 180 ^ { \circ } + i \sin 180 ^ { \circ } \right)$

Accepted Answer

A)  9 
B)  $- 9 \mathrm { i }$ 
C)  $9 \mathrm { i }$ 
D)  $- 9$ 
A)  9 
B)  $- 9 \mathrm { i }$ 
C)  $9 \mathrm { i }$ 
D)  $- 9$

Question 7

Test the equation for symmetry with respect to the given axis, line, or pole.
-$\mathrm { r } = 3 - 6 \sin \theta$; the polar axis

Accepted Answer

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

Question 8

Transform the polar equation to an equation in rectangular coordinates. Then identify and graph the equation.
-$$r = 6 \sin \theta$$

Accepted Answer

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

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

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

Question 9

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

Accepted Answer

A)  $5 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$ 
B)  $\frac { 8 } { 3 } \left( \cos \frac { \pi } { 12 } + i \sin \frac { \pi } { 12 } \right)$ 
C)  $5 \left( \cos \frac { \pi } { 12 } + i \sin \frac { \pi } { 12 } \right)$ 
D)  $\frac { 8 } { 3 } \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$ 
A)  $5 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$ 
B)  $\frac { 8 } { 3 } \left( \cos \frac { \pi } { 12 } + i \sin \frac { \pi } { 12 } \right)$ 
C)  $5 \left( \cos \frac { \pi } { 12 } + i \sin \frac { \pi } { 12 } \right)$ 
D)  $\frac { 8 } { 3 } \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$

Question 10

Find the indicated cross product.
-$$\mathbf { v } = - 4 \mathbf { i } - 4 \mathbf { j } + 4 \mathbf { k } , \mathbf { w } = - 4 \mathbf { i } + 2 \mathbf { k }$$
Find $\mathbf { w } \times \mathbf { v }$.

Accepted Answer

A)  $16 i - 8 j - 16 k$ 
B)  $16 i - 8 j + 8 k$ 
C)  $- 8 \mathbf { i } - 8 \mathbf { j } + 16 \mathbf { k }$ 
D)  $8 i + 8 j + 16 k$ 
A)  $16 i - 8 j - 16 k$ 
B)  $16 i - 8 j + 8 k$ 
C)  $- 8 \mathbf { i } - 8 \mathbf { j } + 16 \mathbf { k }$ 
D)  $8 i + 8 j + 16 k$

Question 11

Find the indicated cross product.
-$$v = 6 i - 5 j + 4 k , \quad w = - 6 i + 2 j + 3 k$$
Find $\mathbf { v } \times \mathbf { w }$.

Accepted Answer

A)  $- 23 \mathbf { i } - 42 \mathrm { j } - 18 \mathbf { k }$ 
B)  $- 18 \mathbf { i } + 23 \mathbf { j } + 42 \mathbf { k }$ 
C)  $- 26 \mathbf { i } - 42 \mathbf { j } - 18 \mathbf { k }$ 
D)  $- 7 \mathbf { i } + 6 \mathbf { j } + 42 \mathbf { k }$ 
A)  $- 23 \mathbf { i } - 42 \mathrm { j } - 18 \mathbf { k }$ 
B)  $- 18 \mathbf { i } + 23 \mathbf { j } + 42 \mathbf { k }$ 
C)  $- 26 \mathbf { i } - 42 \mathbf { j } - 18 \mathbf { k }$ 
D)  $- 7 \mathbf { i } + 6 \mathbf { j } + 42 \mathbf { k }$

Question 12

Find the angle between v and w. Round your answer to one decimal place, if necessary.
-$\mathbf { v } = - 4 \mathbf { i } , \quad \mathrm { w } = \mathbf { j }$

Accepted Answer

A)  $0 ^ { \circ }$ 
B)  $270 ^ { \circ }$ 
C)  $90 ^ { \circ }$ 
D)  $180 ^ { \circ }$ 
A)  $0 ^ { \circ }$ 
B)  $270 ^ { \circ }$ 
C)  $90 ^ { \circ }$ 
D)  $180 ^ { \circ }$

Question 13

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 14

Choose the one alternative that best completes the statement or answers the question.
The polar coordinates of a point are given. Find the rectangular coordinates of the point.
-$$\left( - 3 , - 135 ^ { \circ } \right)$$

Accepted Answer

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

Question 15

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

Accepted Answer

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

Question 16

Find the distance from P1 to P2.
-$$P _ { 1 } = ( - 3,1,1 ) \text { and } P _ { 2 } = ( 1,0 , - 3 )$$

Accepted Answer

A)  $4 \sqrt { 2 }$ 
B)  $\sqrt { 33 }$ 
C)  $\sqrt { 31 }$ 
D)  $\sqrt { 17 }$ 
A)  $4 \sqrt { 2 }$ 
B)  $\sqrt { 33 }$ 
C)  $\sqrt { 31 }$ 
D)  $\sqrt { 17 }$

Question 17

Choose the one alternative that best completes the statement or answers the question.
Find the requested vector.
-$$v = 4 i + 4 j + k$$
Find a vector orthogonal to both $\mathbf { v }$ and $\mathbf { i } + \mathbf { k }$.

Accepted Answer

A)  $- 3 i + 5 j + 3 k$ 
B)  $- 3 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k }$ 
C)  $4 \mathbf { i } - 3 \mathbf { j } - 4 \mathbf { k }$ 
D)  $\mathbf { i } - \mathbf { j } + 3 \mathbf { k }$ 
A)  $- 3 i + 5 j + 3 k$ 
B)  $- 3 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k }$ 
C)  $4 \mathbf { i } - 3 \mathbf { j } - 4 \mathbf { k }$ 
D)  $\mathbf { i } - \mathbf { j } + 3 \mathbf { k }$

Question 18

Choose the one alternative that best completes the statement or answers the question.
-A plane is headed due south with an airspeed of 210 miles per hour. A wind from a direction of $\mathrm { S } 30 ^ { \circ } \mathrm { W }$ is blowing at 20 miles per hour. Find the groundspeed and resulting direction of the plane, rounded to the nearest whole number.

Accepted Answer

A)  $211 \mathrm { mph } ; \mathrm { S } 15 ^ { \circ } \mathrm { W }$ 
B)  $201 \mathrm { mph } ; \mathrm { S } ^ { \circ } \mathrm { E }$ 
C)  $193 \mathrm { mph } ; \mathrm { S } ^ { \circ } \mathrm { E }$ 
D)  $193 \mathrm { mph } ; \mathrm { S } ^ { \circ } \mathrm { W }$ 
A)  $211 \mathrm { mph } ; \mathrm { S } 15 ^ { \circ } \mathrm { W }$ 
B)  $201 \mathrm { mph } ; \mathrm { S } ^ { \circ } \mathrm { E }$ 
C)  $193 \mathrm { mph } ; \mathrm { S } ^ { \circ } \mathrm { E }$ 
D)  $193 \mathrm { mph } ; \mathrm { S } ^ { \circ } \mathrm { W }$

Question 19

Choose the one alternative that best completes the statement or answers the question.
-figure. An audio speaker that weighs 50 pounds hangs from the ceiling of a restaurant from two cables as shown in the 1 To two decimal places, what is the tension in the two cables?

Accepted Answer

A)  Tension in right cable: $35.90 \mathrm { lb }$; tension in left cable: $14.10 \mathrm { lb }$ 
B)  Tension in right cable: $14.10 \mathrm { lb }$; tension in left cable: $41.59 \mathrm { lb }$ 
C)  Tension in right cable: $35.90 \mathrm { lb }$; tension in left cable: $41.59 \mathrm { lb }$ 
D)  Tension in right cable: $41.59 \mathrm { lb }$; tension in left cable: $35.90 \mathrm { lb }$ 
A)  Tension in right cable: $35.90 \mathrm { lb }$; tension in left cable: $14.10 \mathrm { lb }$ 
B)  Tension in right cable: $14.10 \mathrm { lb }$; tension in left cable: $41.59 \mathrm { lb }$ 
C)  Tension in right cable: $35.90 \mathrm { lb }$; tension in left cable: $41.59 \mathrm { lb }$ 
D)  Tension in right cable: $41.59 \mathrm { lb }$; tension in left cable: $35.90 \mathrm { lb }$

Question 20

Plot the point given in polar coordinates.
-$$\left( 2 , \frac { 9 \pi } { 4 } \right)$$

Accepted Answer

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

The rectangular coordinates of a point are given. Find polar coordinates for the point. - $( 0,9 )$

Identify and graph the polar equation. - $r = 5 - 4 \cos \theta$ $Identify and graph the polar equation. - r = 5 - 4 \cos \theta$

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

The vector v has initial position P and terminal point Q. Write v in the form ai + bj; that is, find its position vector. - $\mathrm { P } = ( 0,0 ) ; \mathrm { Q } = ( 6,4 )$

Choose the one alternative that best completes the statement or answers the question. -Which of the following vectors is parallel to $\mathbf { v } = 9 \mathbf { i } + 5 \mathbf { j }$ ?

Write the complex number in rectangular form. - $9 \left( \cos 180 ^ { \circ } + i \sin 180 ^ { \circ } \right)$

Test the equation for symmetry with respect to the given axis, line, or pole. - $\mathrm { r } = 3 - 6 \sin \theta$ ; the polar axis

Transform the polar equation to an equation in rectangular coordinates. Then identify and graph the equation. - $r = 6 \sin \theta$ $Transform the polar equation to an equation in rectangular coordinates. Then identify and graph the equation. - r = 6 \sin \theta$

Solve the problem. Leave your answer in polar form. - z=8 +i w=3 +i Find $\frac { z } { w }$ .

Find the indicated cross product. - $\mathbf { v } = - 4 \mathbf { i } - 4 \mathbf { j } + 4 \mathbf { k } , \mathbf { w } = - 4 \mathbf { i } + 2 \mathbf { k }$ Find $\mathbf { w } \times \mathbf { v }$ .

Find the indicated cross product. - $v = 6 i - 5 j + 4 k , \quad w = - 6 i + 2 j + 3 k$ Find $\mathbf { v } \times \mathbf { w }$ .

Find the angle between v and w. Round your answer to one decimal place, if necessary. - $\mathbf { v } = - 4 \mathbf { i } , \quad \mathrm { w } = \mathbf { j }$

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

Choose the one alternative that best completes the statement or answers the question. The polar coordinates of a point are given. Find the rectangular coordinates of the point. - $\left( - 3 , - 135 ^ { \circ } \right)$

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

Find the distance from P1 to P2. - $P _ { 1 } = ( - 3,1,1 ) \text { and } P _ { 2 } = ( 1,0 , - 3 )$

Choose the one alternative that best completes the statement or answers the question. Find the requested vector. - $v = 4 i + 4 j + k$ Find a vector orthogonal to both $\mathbf { v }$ and $\mathbf { i } + \mathbf { k }$ .

Choose the one alternative that best completes the statement or answers the question. -figure. An audio speaker that weighs 50 pounds hangs from the ceiling of a restaurant from two cables as shown in the 1 To two decimal places, what is the tension in the two cables?

Plot the point given in polar coordinates. - $\left( 2 , \frac { 9 \pi } { 4 } \right)$ $Plot the point given in polar coordinates. - \left( 2 , \frac { 9 \pi } { 4 } \right)$

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

Foundations: a Prelude to Functions

Graphing Utilities

Filters

Exam 8: Polar Coordinates; Vectors

The rectangular coordinates of a point are given. Find polar coordinates for the point. - (0,9)( 0,9 )(0,9)

Identify and graph the polar equation. - r=5−4cos⁡θr = 5 - 4 \cos \thetar=5−4cosθ

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

The vector v has initial position P and terminal point Q. Write v in the form ai + bj; that is, find its position vector. - P=(0,0);Q=(6,4)\mathrm { P } = ( 0,0 ) ; \mathrm { Q } = ( 6,4 )P=(0,0);Q=(6,4)

Choose the one alternative that best completes the statement or answers the question. -Which of the following vectors is parallel to v=9i+5j\mathbf { v } = 9 \mathbf { i } + 5 \mathbf { j }v=9i+5j ?

Write the complex number in rectangular form. - 9(cos⁡180∘+isin⁡180∘)9 \left( \cos 180 ^ { \circ } + i \sin 180 ^ { \circ } \right)9(cos180∘+isin180∘)

Test the equation for symmetry with respect to the given axis, line, or pole. - r=3−6sin⁡θ\mathrm { r } = 3 - 6 \sin \thetar=3−6sinθ ; the polar axis

Transform the polar equation to an equation in rectangular coordinates. Then identify and graph the equation. - r=6sin⁡θr = 6 \sin \thetar=6sinθ

Solve the problem. Leave your answer in polar form. - z=8 +i w=3 +i Find zw\frac { z } { w }wz​ .

Find the indicated cross product. - v=−4i−4j+4k,w=−4i+2k\mathbf { v } = - 4 \mathbf { i } - 4 \mathbf { j } + 4 \mathbf { k } , \mathbf { w } = - 4 \mathbf { i } + 2 \mathbf { k }v=−4i−4j+4k,w=−4i+2k Find w×v\mathbf { w } \times \mathbf { v }w×v .

Find the indicated cross product. - v=6i−5j+4k,w=−6i+2j+3kv = 6 i - 5 j + 4 k , \quad w = - 6 i + 2 j + 3 kv=6i−5j+4k,w=−6i+2j+3k Find v×w\mathbf { v } \times \mathbf { w }v×w .

Find the angle between v and w. Round your answer to one decimal place, if necessary. - v=−4i,w=j\mathbf { v } = - 4 \mathbf { i } , \quad \mathrm { w } = \mathbf { j }v=−4i,w=j