Exam 9: Vectors in Two and Three Dimensions

Let $z _ { 1 } = 8 \left( \cos \frac { 11 \pi } { 6 } + i \sin \frac { 11 \pi } { 6 } \right)$ and $z _ { 2 } = 2 \sqrt { 3 } \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$ ) Find $z _ { 1 } / z _ { 2 }$

(Multiple Choice)

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Question 21

Convert the equation to polar form. $x ^ { 2 } + y ^ { 2 } = 25$

(Short Answer)

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Question 22

Convert the equation to polar form. $x ^ { 2 } - y ^ { 2 } = 4$

(Multiple Choice)

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Question 23

Sketch a graph of the rectangular equation. [Hint: First convert the equation to polar coordinates.] $\left( x ^ { 2 } + y ^ { 2 } + 3 y \right) ^ { 2 } = 9 \left( x ^ { 2 } + y ^ { 2 } \right)$

(Multiple Choice)

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Question 24

Find two polar coordinate representations for the point $( 3 , \pi / 3 )$ , one with $r> 0$ , and the other with $r < 0$

(Short Answer)

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Question 25

Convert the equation to polar form. $x ^ { 2 } + y ^ { 2 } = 16$

(Short Answer)

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Question 26

Write $z _ { 1 } = 2 + 2 i$ in polar form then find $1 / z _ { 1 }$

(Multiple Choice)

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Question 27

Find a rectangular-coordinate equation for the curve by eliminating the parameter. $x = 4 - t ^ { 2 } , y = 2 + t$

(Short Answer)

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Question 28

Convert the rectangular coordinates to polar coordinates with $r > 0 \text { and } 0 \leq \theta < 2 \pi$ $( - 2 \sqrt { 3 } , - 2 )$

(Short Answer)

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Question 29

Find a rectangular-coordinate equation for the curve by eliminating the parameter. $x = t + 3 , y = \frac { t } { t + 3 }$

(Multiple Choice)

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Question 30

Convert the rectangular coordinates to polar coordinates with $r > 0 \text { and } 0 \leq \theta < 2 \pi$ $( - \sqrt { 3 } , - 1 )$

(Short Answer)

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Question 31

Find the modulus and the argument for the complex number. $z = - i$

(Multiple Choice)

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Question 32

Write the complex conjugate of z in polar form with argument $\theta$ between 0 and $2 \pi$ $z = - 5 - 5 i \sqrt { 3 }$

(Short Answer)

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Question 33

Write the complex conjugate of z in polar form with argument $\theta$ between 0 and $2 \pi$ $z = 5 - 5 i \sqrt { 3 }$

(Short Answer)

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Question 34

Sketch a graph of the polar equation. $y = \sqrt { 3 } + 2 \sin \theta$

(Essay)

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Question 35

Find a rectangular-coordinate equation for the curve by eliminating the parameter. $x = t + 2 , y = \frac { t } { t + 2 }$

(Multiple Choice)

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Question 36

Find two polar coordinate representations for the point $( 3 , \pi / 3 )$ , one with $r > 0$ , and both with $0 \leq \theta < 2 \pi$

(Short Answer)

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Question 37

Graph the polar equation $r= 4 \sin \theta$

(Essay)

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Question 38

Use DeMoivre's Theorem to find the indicated power. $(1+i)^{2}$

(Short Answer)

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Question 39

Sketch a graph of the polar equation. $r=\sqrt{3}-\cos \theta$

(Multiple Choice)

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Question 40

Let $z _ { 1 } = 8 \left( \cos \frac { 11 \pi } { 6 } + i \sin \frac { 11 \pi } { 6 } \right)$ and $z _ { 2 } = 2 \sqrt { 3 } \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$ ) Find $z _ { 1 } / z _ { 2 }$

Convert the equation to polar form. $x ^ { 2 } + y ^ { 2 } = 25$

Convert the equation to polar form. $x ^ { 2 } - y ^ { 2 } = 4$

Sketch a graph of the rectangular equation. [Hint: First convert the equation to polar coordinates.] $\left( x ^ { 2 } + y ^ { 2 } + 3 y \right) ^ { 2 } = 9 \left( x ^ { 2 } + y ^ { 2 } \right)$

Find two polar coordinate representations for the point $( 3 , \pi / 3 )$ , one with $r> 0$ , and the other with $r < 0$

Convert the equation to polar form. $x ^ { 2 } + y ^ { 2 } = 16$

Write $z _ { 1 } = 2 + 2 i$ in polar form then find $1 / z _ { 1 }$

Find a rectangular-coordinate equation for the curve by eliminating the parameter. $x = 4 - t ^ { 2 } , y = 2 + t$

Convert the rectangular coordinates to polar coordinates with $r > 0 \text { and } 0 \leq \theta < 2 \pi$ $( - 2 \sqrt { 3 } , - 2 )$

Find a rectangular-coordinate equation for the curve by eliminating the parameter. $x = t + 3 , y = \frac { t } { t + 3 }$

Convert the rectangular coordinates to polar coordinates with $r > 0 \text { and } 0 \leq \theta < 2 \pi$ $( - \sqrt { 3 } , - 1 )$

Find the modulus and the argument for the complex number. $z = - i$

Write the complex conjugate of z in polar form with argument $\theta$ between 0 and $2 \pi$ $z = - 5 - 5 i \sqrt { 3 }$

Write the complex conjugate of z in polar form with argument $\theta$ between 0 and $2 \pi$ $z = 5 - 5 i \sqrt { 3 }$

Sketch a graph of the polar equation. $y = \sqrt { 3 } + 2 \sin \theta$

Find a rectangular-coordinate equation for the curve by eliminating the parameter. $x = t + 2 , y = \frac { t } { t + 2 }$

Find two polar coordinate representations for the point $( 3 , \pi / 3 )$ , one with $r > 0$ , and both with $0 \leq \theta < 2 \pi$

Graph the polar equation $r= 4 \sin \theta$

Use DeMoivre's Theorem to find the indicated power. $(1+i)^{2}$

Sketch a graph of the polar equation. $r=\sqrt{3}-\cos \theta$

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Trigonometric Functions: Right Triangle Approach

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Exam 9: Vectors in Two and Three Dimensions

Convert the equation to polar form. x2+y2=25x ^ { 2 } + y ^ { 2 } = 25x2+y2=25

Convert the equation to polar form. x2−y2=4x ^ { 2 } - y ^ { 2 } = 4x2−y2=4

Sketch a graph of the rectangular equation. [Hint: First convert the equation to polar coordinates.] (x2+y2+3y)2=9(x2+y2)\left( x ^ { 2 } + y ^ { 2 } + 3 y \right) ^ { 2 } = 9 \left( x ^ { 2 } + y ^ { 2 } \right)(x2+y2+3y)2=9(x2+y2)

Find two polar coordinate representations for the point (3,π/3)( 3 , \pi / 3 )(3,π/3) , one with r>0r> 0r>0 , and the other with r<0r < 0r<0

Convert the equation to polar form. x2+y2=16x ^ { 2 } + y ^ { 2 } = 16x2+y2=16

Write z1=2+2iz _ { 1 } = 2 + 2 iz1​=2+2i in polar form then find 1/z11 / z _ { 1 }1/z1​

Find a rectangular-coordinate equation for the curve by eliminating the parameter. x=4−t2,y=2+tx = 4 - t ^ { 2 } , y = 2 + tx=4−t2,y=2+t

Convert the rectangular coordinates to polar coordinates with r>0 and 0≤θ<2πr > 0 \text { and } 0 \leq \theta < 2 \pir>0 and 0≤θ<2π (−23,−2)( - 2 \sqrt { 3 } , - 2 )(−23​,−2)

Find a rectangular-coordinate equation for the curve by eliminating the parameter. x=t+3,y=tt+3x = t + 3 , y = \frac { t } { t + 3 }x=t+3,y=t+3t​

Convert the rectangular coordinates to polar coordinates with r>0 and 0≤θ<2πr > 0 \text { and } 0 \leq \theta < 2 \pir>0 and 0≤θ<2π (−3,−1)( - \sqrt { 3 } , - 1 )(−3​,−1)

Find the modulus and the argument for the complex number. z=−iz = - iz=−i

Write the complex conjugate of z in polar form with argument θ\thetaθ between 0 and 2π2 \pi2π z=−5−5i3z = - 5 - 5 i \sqrt { 3 }z=−5−5i3​

Write the complex conjugate of z in polar form with argument θ\thetaθ between 0 and 2π2 \pi2π z=5−5i3z = 5 - 5 i \sqrt { 3 }z=5−5i3​

Sketch a graph of the polar equation. y=3+2sin⁡θy = \sqrt { 3 } + 2 \sin \thetay=3​+2sinθ

Find a rectangular-coordinate equation for the curve by eliminating the parameter. x=t+2,y=tt+2x = t + 2 , y = \frac { t } { t + 2 }x=t+2,y=t+2t​

Find two polar coordinate representations for the point (3,π/3)( 3 , \pi / 3 )(3,π/3) , one with r>0r > 0r>0 , and both with 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π

Graph the polar equation r=4sin⁡θr= 4 \sin \thetar=4sinθ

Use DeMoivre's Theorem to find the indicated power. (1+i)2(1+i)^{2}(1+i)2

Sketch a graph of the polar equation. r=3−cos⁡θr=\sqrt{3}-\cos \thetar=3​−cosθ

Equations and Graphs

Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions: Right Triangle Approach

Trigonometric Functions: Unit Circle Approach

Conic Sections

Polar Coordinates and Parametric Equations

Systems of Equations and Inequalities

Matrices and Determinants

Conic Sections

Sequences and Series

Counting and Probability

Filters

Convert the equation to polar form. $x ^ { 2 } + y ^ { 2 } = 25$

Convert the equation to polar form. $x ^ { 2 } - y ^ { 2 } = 4$

Sketch a graph of the rectangular equation. [Hint: First convert the equation to polar coordinates.] $\left( x ^ { 2 } + y ^ { 2 } + 3 y \right) ^ { 2 } = 9 \left( x ^ { 2 } + y ^ { 2 } \right)$

Find two polar coordinate representations for the point $( 3 , \pi / 3 )$ , one with $r> 0$ , and the other with $r < 0$

Convert the equation to polar form. $x ^ { 2 } + y ^ { 2 } = 16$

Write $z _ { 1 } = 2 + 2 i$ in polar form then find $1 / z _ { 1 }$

Find a rectangular-coordinate equation for the curve by eliminating the parameter. $x = 4 - t ^ { 2 } , y = 2 + t$

Convert the rectangular coordinates to polar coordinates with $r > 0 \text { and } 0 \leq \theta < 2 \pi$ $( - 2 \sqrt { 3 } , - 2 )$

Find a rectangular-coordinate equation for the curve by eliminating the parameter. $x = t + 3 , y = \frac { t } { t + 3 }$

Convert the rectangular coordinates to polar coordinates with $r > 0 \text { and } 0 \leq \theta < 2 \pi$ $( - \sqrt { 3 } , - 1 )$

Find the modulus and the argument for the complex number. $z = - i$

Write the complex conjugate of z in polar form with argument $\theta$ between 0 and $2 \pi$ $z = - 5 - 5 i \sqrt { 3 }$

Write the complex conjugate of z in polar form with argument $\theta$ between 0 and $2 \pi$ $z = 5 - 5 i \sqrt { 3 }$

Sketch a graph of the polar equation. $y = \sqrt { 3 } + 2 \sin \theta$

Find a rectangular-coordinate equation for the curve by eliminating the parameter. $x = t + 2 , y = \frac { t } { t + 2 }$

Find two polar coordinate representations for the point $( 3 , \pi / 3 )$ , one with $r > 0$ , and both with $0 \leq \theta < 2 \pi$

Graph the polar equation $r= 4 \sin \theta$

Use DeMoivre's Theorem to find the indicated power. $(1+i)^{2}$

Sketch a graph of the polar equation. $r=\sqrt{3}-\cos \theta$