Exam 9: Vectors in Two and Three Dimensions

Let $z _ { 1 } = 2 \left( \cos \frac { 5 \pi } { 6 } + i \sin \frac { 5 \pi } { 6 } \right)$ and $z _ { 2 } = 5 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$ . Find $Z _ { 1 } Z _ { 2 }$

Find the modulus and the argument for the complex number. $z = - 15$

Which of the following is not a polar point representation for the point $( 3 , \pi / 3 )$ ?

Convert the equation to polar form. $x ^ { 2 } - y ^ { 2 } = 9$

Let $z _ { 1 } = 4 \left( \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \right)$ and $z _ { 2 } = 5 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$ . Find $Z _ { 1 } Z _ { 2 }$

Use DeMoivre's Theorem to find the indicated power. $( 1 + i ) ^ { 20 }$

If a projectile is fired with an initial speed of $v _ { 0 }$ ft/s at an angle $\alpha$ above the horizontal, then its position after t seconds is given by the parametric equations $x = \left( v _ { 0 } \cos \alpha \right) t \text { and } y = \left( v _ { 0 } \sin \alpha \right) t - 16 t ^ { 2 }$ where x and y are measured in feet.Suppose a gun fires a bullet into the air with an initial speed of 1024 ft/s at an angle of $30 ^ { \circ }$ to the horizontal. After how many seconds will the bullet hit the ground?

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line $\theta = \pi / 2$ . $r = 3 \cos \theta \csc \theta$

Let $z _ { 1 } = 2 \left( \cos \frac { 5 \pi } { 6 } + i \sin \frac { 5 \pi } { 6 } \right)$ and $z _ { 2 } = 5 \left( \cos\frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$ . Find $Z _ { 1 } Z _ { 2 }$

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

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

Convert the polar equation to rectangular coordinates. $\frac {r } { 6 } = \sec \theta$

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

Find the rectangular-coordinate equation for the parametric equations given. $x = 2 \cos ^ { 2 } t , y = 2 \sin ^ { 2 } t$

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

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

Convert the polar equation to rectangular coordinates. $\frac {r } { 6 } = \sec \theta$

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

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

Convert the polar equation to rectangular coordinates. $\frac { r } { 6 } = \sec \theta$