Exam 9: Vectors in Two and Three Dimensions

Find parametric equations for the line with the given properties.Passing through $( 12,4 )$ and the origin

Sketch a graph of the rectangular equation. [Hint: First convert the equation to polar coordinates.] $\left( x ^ { 2 } + y ^ { 2 } + x \right) ^ { 2 } = 16 \left( x ^ { 2 } + y ^ { 2 } \right)$

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line $\theta = \pi / 2$ . $r ^ { 2 } = 9 \cos 2 \theta$

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

Sketch the curve represented by the parametric equations and find its rectangular-coordinate equation. $x = 2 \cos ^ { 2 } t , y = 2 \sin ^ { 2 } t$

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

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

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

Convert the polar equation to rectangular coordinates. $r + \cos \theta = 3$

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

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. How far from the gun will the bullet hit the ground?

Solve the equation. $x ^ { 4 } + 1 = 0$

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

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 $Z _ { 1 } Z _ { 2 }$

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?