Solve the problem. Leave your answer in polar form. -\[\begin{array} { l } \left. z = 6 \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right) \\ w = 12 \left( \cos \frac { 5 \pi } { 6 } + i \sin \frac { 5 \pi } { 6 } \right) \end{array}\] Find zw.

A) $36 \left( \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \right)$ B) $72 \left( \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \right)$ C) $\left. 36 \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$ D) $72 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$ A) $36 \left( \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \right)$ B) $72 \left( \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \right)$ C) $\left. 36 \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$ D) $72 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$

Exam 8: Polar Coordinates; Vectors

State whether the vectors are parallel, orthogonal, or neither. - $\mathbf { v } = 4 \mathbf { i } + 2 \mathbf { j } , \quad \mathbf { w } = 8 \mathbf { i } + 4 \mathbf { j }$

(Multiple Choice)

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Question 221

Find all the complex roots. Leave your answers in polar form with the argument in degrees. -The complex fourth roots of $- 16$

(Multiple Choice)

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Question 222

The vector v has initial position P and terminal point Q. Write v in the form ai + bj; that is, find its position vector. - $\mathrm { P } = ( 2 , - 6 ) ; \mathrm { Q } = ( - 3,6 )$

(Multiple Choice)

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Question 223

Describe the set of points (x, y, z) defined by the equation. -x = -2

(Multiple Choice)

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Question 224

Solve the problem. Leave your answer in polar form. - z=6+i w=12 +i Find zw.

(Multiple Choice)

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Question 225

Decompose v into two vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w. - $\mathbf { v } = \mathbf { i } + 5 \mathbf { j } , \quad \mathbf { w } = - 2 \mathbf { i } + \mathbf { j }$

(Multiple Choice)

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Question 226

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

(Multiple Choice)

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Question 227

State whether the vectors are parallel, orthogonal, or neither. - $\mathbf { v } = \mathrm { i } + \sqrt { 3 } \mathbf { j } , \quad \mathbf { w } = \mathrm { i } - 2 \mathbf { j }$

(Multiple Choice)

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Question 228

Graph the polar equation. - $\mathrm{r}=2-2 \cos \theta$ $Graph the polar equation. - \mathrm{r}=2-2 \cos \theta$

(Multiple Choice)

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Question 229

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. - $\left( - 9 , \frac { 2 \pi } { 3 } \right)$

(Multiple Choice)

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Question 230

Graph the polar equation. - $r=\sin \theta \tan \theta$ $Graph the polar equation. - r=\sin \theta \tan \theta$

(Multiple Choice)

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Question 231

Solve the problem. -Find a so that the vectors $\mathbf { v } = \mathbf { i } - \mathrm { a } \mathbf { j }$ and $\mathbf { w } = - 5 \mathbf { i } - 3 \mathbf { j }$ are orthogonal.

(Multiple Choice)

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Question 232

Solve the problem. -If $\mathbf { u } = - 9 i - 2 \mathbf { j }$ and $\mathbf { v } = - 3 \mathbf { i } + 7 \mathbf { j }$ , find $\mathbf { u } - \mathbf { v }$ .

(Multiple Choice)

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Question 233

Solve the problem. Round your answer to the nearest tenth. -A wagon is pulled horizontally by exerting a force of 60 pounds on the handle at an angle of 25° to the horizontal. How much work is done in moving the wagon 50 feet?" .

(Multiple Choice)

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Question 234

Find the dot product v · w. - $\mathbf { v } = 2 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }$ and $\mathbf { w } = 3 \mathbf { i } + 7 \mathbf { j } + 3 \mathbf { k }$

(Multiple Choice)

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Question 235

Find the value of the determinant. - $\left| \begin{array} { l l } 8 & 5 \\7 & 7\end{array} \right|$

(Multiple Choice)

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Question 236

Decompose v into two vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w. - $\mathbf { v } = - 3 \mathbf { i } - 5 \mathbf { j } , \quad \mathbf { w } = 2 \mathbf { i } - \mathbf { j }$

(Multiple Choice)

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Question 237

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

(Multiple Choice)

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Question 238

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

(Multiple Choice)

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Question 239

The rectangular coordinates of a point are given. Find polar coordinates for the point. - $( - 2,0.6 )$ Round the polar coordinates to two decimal places, with $\theta$ in radians.

(Multiple Choice)

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Question 240

State whether the vectors are parallel, orthogonal, or neither. - $\mathbf { v } = 4 \mathbf { i } + 2 \mathbf { j } , \quad \mathbf { w } = 8 \mathbf { i } + 4 \mathbf { j }$

Find all the complex roots. Leave your answers in polar form with the argument in degrees. -The complex fourth roots of $- 16$

The vector v has initial position P and terminal point Q. Write v in the form ai + bj; that is, find its position vector. - $\mathrm { P } = ( 2 , - 6 ) ; \mathrm { Q } = ( - 3,6 )$

Describe the set of points (x, y, z) defined by the equation. -x = -2

Solve the problem. Leave your answer in polar form. - z=6+i w=12 +i Find zw.

Decompose v into two vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w. - $\mathbf { v } = \mathbf { i } + 5 \mathbf { j } , \quad \mathbf { w } = - 2 \mathbf { i } + \mathbf { j }$

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

State whether the vectors are parallel, orthogonal, or neither. - $\mathbf { v } = \mathrm { i } + \sqrt { 3 } \mathbf { j } , \quad \mathbf { w } = \mathrm { i } - 2 \mathbf { j }$

Graph the polar equation. - $\mathrm{r}=2-2 \cos \theta$ $Graph the polar equation. - \mathrm{r}=2-2 \cos \theta$

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. - $\left( - 9 , \frac { 2 \pi } { 3 } \right)$

Graph the polar equation. - $r=\sin \theta \tan \theta$ $Graph the polar equation. - r=\sin \theta \tan \theta$

Solve the problem. -Find a so that the vectors $\mathbf { v } = \mathbf { i } - \mathrm { a } \mathbf { j }$ and $\mathbf { w } = - 5 \mathbf { i } - 3 \mathbf { j }$ are orthogonal.

Solve the problem. -If $\mathbf { u } = - 9 i - 2 \mathbf { j }$ and $\mathbf { v } = - 3 \mathbf { i } + 7 \mathbf { j }$ , find $\mathbf { u } - \mathbf { v }$ .

Solve the problem. Round your answer to the nearest tenth. -A wagon is pulled horizontally by exerting a force of 60 pounds on the handle at an angle of 25° to the horizontal. How much work is done in moving the wagon 50 feet?" .

Find the dot product v · w. - $\mathbf { v } = 2 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }$ and $\mathbf { w } = 3 \mathbf { i } + 7 \mathbf { j } + 3 \mathbf { k }$

Find the value of the determinant. - $\left| \begin{array} { l l } 8 & 5 \\7 & 7\end{array} \right|$

Decompose v into two vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w. - $\mathbf { v } = - 3 \mathbf { i } - 5 \mathbf { j } , \quad \mathbf { w } = 2 \mathbf { i } - \mathbf { j }$

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

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

The rectangular coordinates of a point are given. Find polar coordinates for the point. - $( - 2,0.6 )$ Round the polar coordinates to two decimal places, with $\theta$ in radians.

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

State whether the vectors are parallel, orthogonal, or neither. - v=4i+2j,w=8i+4j\mathbf { v } = 4 \mathbf { i } + 2 \mathbf { j } , \quad \mathbf { w } = 8 \mathbf { i } + 4 \mathbf { j }v=4i+2j,w=8i+4j

Find all the complex roots. Leave your answers in polar form with the argument in degrees. -The complex fourth roots of −16- 16−16

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=(2,−6);Q=(−3,6)\mathrm { P } = ( 2 , - 6 ) ; \mathrm { Q } = ( - 3,6 )P=(2,−6);Q=(−3,6)

Describe the set of points (x, y, z) defined by the equation. -x = -2

Solve the problem. Leave your answer in polar form. - z=6+i w=12 +i Find zw.

Decompose v into two vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w. - v=i+5j,w=−2i+j\mathbf { v } = \mathbf { i } + 5 \mathbf { j } , \quad \mathbf { w } = - 2 \mathbf { i } + \mathbf { j }v=i+5j,w=−2i+j

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

State whether the vectors are parallel, orthogonal, or neither. - v=i+3j,w=i−2j\mathbf { v } = \mathrm { i } + \sqrt { 3 } \mathbf { j } , \quad \mathbf { w } = \mathrm { i } - 2 \mathbf { j }v=i+3​j,w=i−2j

Graph the polar equation. - r=2−2cos⁡θ\mathrm{r}=2-2 \cos \thetar=2−2cosθ

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. - (−9,2π3)\left( - 9 , \frac { 2 \pi } { 3 } \right)(−9,32π​)

Graph the polar equation. - r=sin⁡θtan⁡θr=\sin \theta \tan \thetar=sinθtanθ

Solve the problem. -Find a so that the vectors v=i−aj\mathbf { v } = \mathbf { i } - \mathrm { a } \mathbf { j }v=i−aj and w=−5i−3j\mathbf { w } = - 5 \mathbf { i } - 3 \mathbf { j }w=−5i−3j are orthogonal.

Solve the problem. -If u=−9i−2j\mathbf { u } = - 9 i - 2 \mathbf { j }u=−9i−2j and v=−3i+7j\mathbf { v } = - 3 \mathbf { i } + 7 \mathbf { j }v=−3i+7j , find u−v\mathbf { u } - \mathbf { v }u−v .

Solve the problem. Round your answer to the nearest tenth. -A wagon is pulled horizontally by exerting a force of 60 pounds on the handle at an angle of 25° to the horizontal. How much work is done in moving the wagon 50 feet?" .

Find the dot product v · w. - v=2i+3j+k\mathbf { v } = 2 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }v=2i+3j+k and w=3i+7j+3k\mathbf { w } = 3 \mathbf { i } + 7 \mathbf { j } + 3 \mathbf { k }w=3i+7j+3k

Find the value of the determinant. - ∣8577∣\left| \begin{array} { l l } 8 & 5 \\7 & 7\end{array} \right|​87​57​​

Decompose v into two vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w. - v=−3i−5j,w=2i−j\mathbf { v } = - 3 \mathbf { i } - 5 \mathbf { j } , \quad \mathbf { w } = 2 \mathbf { i } - \mathbf { j }v=−3i−5j,w=2i−j

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

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

The rectangular coordinates of a point are given. Find polar coordinates for the point. - (−2,0.6)( - 2,0.6 )(−2,0.6) Round the polar coordinates to two decimal places, with θ\thetaθ in radians.

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

State whether the vectors are parallel, orthogonal, or neither. - $\mathbf { v } = 4 \mathbf { i } + 2 \mathbf { j } , \quad \mathbf { w } = 8 \mathbf { i } + 4 \mathbf { j }$

Find all the complex roots. Leave your answers in polar form with the argument in degrees. -The complex fourth roots of $- 16$

The vector v has initial position P and terminal point Q. Write v in the form ai + bj; that is, find its position vector. - $\mathrm { P } = ( 2 , - 6 ) ; \mathrm { Q } = ( - 3,6 )$

Decompose v into two vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w. - $\mathbf { v } = \mathbf { i } + 5 \mathbf { j } , \quad \mathbf { w } = - 2 \mathbf { i } + \mathbf { j }$

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

State whether the vectors are parallel, orthogonal, or neither. - $\mathbf { v } = \mathrm { i } + \sqrt { 3 } \mathbf { j } , \quad \mathbf { w } = \mathrm { i } - 2 \mathbf { j }$

Graph the polar equation. - $\mathrm{r}=2-2 \cos \theta$ $Graph the polar equation. - \mathrm{r}=2-2 \cos \theta$

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. - $\left( - 9 , \frac { 2 \pi } { 3 } \right)$

Graph the polar equation. - $r=\sin \theta \tan \theta$ $Graph the polar equation. - r=\sin \theta \tan \theta$

Solve the problem. -Find a so that the vectors $\mathbf { v } = \mathbf { i } - \mathrm { a } \mathbf { j }$ and $\mathbf { w } = - 5 \mathbf { i } - 3 \mathbf { j }$ are orthogonal.

Solve the problem. -If $\mathbf { u } = - 9 i - 2 \mathbf { j }$ and $\mathbf { v } = - 3 \mathbf { i } + 7 \mathbf { j }$ , find $\mathbf { u } - \mathbf { v }$ .

Find the dot product v · w. - $\mathbf { v } = 2 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }$ and $\mathbf { w } = 3 \mathbf { i } + 7 \mathbf { j } + 3 \mathbf { k }$

Find the value of the determinant. - $\left| \begin{array} { l l } 8 & 5 \\7 & 7\end{array} \right|$

Decompose v into two vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w. - $\mathbf { v } = - 3 \mathbf { i } - 5 \mathbf { j } , \quad \mathbf { w } = 2 \mathbf { i } - \mathbf { j }$

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

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

The rectangular coordinates of a point are given. Find polar coordinates for the point. - $( - 2,0.6 )$ Round the polar coordinates to two decimal places, with $\theta$ in radians.