Chapter 9: Polar Coordinates, Vectors

If $z=r(\cos \theta +i\sin \theta )$is a complex number, then

$z^n=\_\_\_\_\_\left[\cos \right(\_\_\_\_)+i\sin (\_\_\_\_\left)\right]$.

If P is a point with polar coordinates $(r,\theta )$the rectangular coordinates $(x,y)$of P are given by x = _____ and y = ______.

In Problems 7–16,

(a) find the dot product $v·w$;

(b) find the angle betweenv andw;

(c) state whether the vectors are parallel, orthogonal, or neither.

$$\begin{gathered} v=i+j \\ w=-i+j \end{gathered}$$

In Problems 8 and 9, test the polar equation for symmetry with respect to the pole, the polar axis, and the line $\theta =\frac{\pi }{2}$.

$r^2\cos \theta =5$

Ground Speed and Direction of an Airplane An airplane has an airspeed of $600km/hr$bearing $S30°E$. The wind velocity is $40km/hr$in the direction $S45°E$. Find the resultant vector representing the path of the plane relative to the ground. What is the ground speed of the plane? What is its direction?

$r=\cos \frac{\theta }{2}$

The letters r and $\theta$represent polar coordinates. Write each equation using rectangular coordinates $(x,y)$.

$r=4$

Weight of a Boat A magnitude of 700 pounds of force is required to hold a boat and its trailer in place on a ramp whose incline is $10°$to the horizontal. What is the combined weight of the boat and its trailer?

Show that the graph of the equation $r\sin \theta =a$is a horizontal

line a units above the pole if a $a\ge 0$and $\left|a\right|$units below the

pole if $a<0$.

The letters r and $\theta$represent polar coordinates. Write each equation using rectangular coordinates $\left(x,y\right)$.

role="math" localid="1646746889027" $r=\frac{4}{1-\cos \theta }$