Exam 9: Vectors in Two and Three Dimensions

Let $z _ { 1 } = \sqrt { 2 } \left( \cos \frac { 7 \pi } { 4 } + i \sin \frac { 7 \pi } { 4 } \right)$ and $z _ { 2 } = 2 \left( \cos \frac { 5 \pi } { 3 } + i \sin \frac { 5 \pi } { 3 } \right)$ . Find $z _ { 1 } / z _ { 2 }$

Find parametric equations for the line with the given properties.Passing through $( 12,8 )$ and the origin

Find parametric equations for the line with the given properties.Passing through $( 12,8 )$ and the origin

Sketch a graph of the polar equation. $r = - 2 \sin 3 \theta$

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

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line $\theta = \pi / 2$ . $r ^ { 2 } = \cos 2 \theta$ I $\text { symmetric about the polar axis }$ II $\text { symmetric about the pole }$ III $\text { symmetric about the line } \theta = \pi / 2$

Solve the equation. $x ^ { 2 } - i = 0$

Convert the polar equation to rectangular coordinates. $\frac { y } { 3 } = \csc \theta$

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

Find parametric equations for the line with the given properties.Passing through $( 12,8 )$ and the origin

Convert the rectangular coordinates to polar coordinates with $r > 0 \text { and } 0 \leq \theta < 2 \pi$ $( 0 , - \sqrt { 5 } )$

Find the cube roots of $i$

Write $z _ { 1 } = 1 - \sqrt { 2 j }$ in polar form then find $1 / z _ { 1 }$

Convert the polar equation to rectangular coordinates. $\frac { y } { 3 } = \csc \theta$

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line $\theta = \pi / 2$ . $r^ { 2 } = \cos 2 \theta$ I $\text { symmetric about the polar axis }$ II $\text { symmetric about the pole }$ III $\text { symmetric about the line } \theta = \pi / 2$

Convert the polar equation to rectangular coordinates. $r + \cos \theta = 4$

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

Graph the polar equation $r = 8 \cos \theta$

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