91Ó°ÊÓ

Convert the point with the given polar coordinates to rectangular coordinates \((x, y) .\) polar coordinates \(\left(10, \frac{\pi}{6}\right)\)

Write each expression in the form \(a+b i,\) where a and b are real numbers. \((5+6 i)(2-7 i)\)

Suppose \(\mathbf{u}=(-3,2)\) and \(\mathbf{v}=(-2,-1)\) (a) Draw a figure using arrows illustrating the difference \(\mathbf{u}-\mathbf{v}\) (b) Compute the difference \(\mathbf{u}-\mathbf{v}\) using coordinates.

Write $$ \left(\cos \frac{\pi}{5}+i \sin \frac{\pi}{5}\right)\left(\cos \frac{\pi}{11}+i \sin \frac{\pi}{11}\right) $$ in polar form.

Convert the point with the given polar coordinates to rectangular coordinates \((x, y) .\) polar coordinates \(\left(11,-\frac{\pi}{6}\right)\)

Suppose the wind at airplane heights is 40 miles per hour (relative to the ground) moving \(15^{\circ}\) north of east. Relative to the wind, an airplane is flying at 450 miles per hour \(20^{\circ}\) south of the wind. Find the speed and direction of the airplane relative to the ground.

Write each expression in the form \(a+b i,\) where a and b are real numbers. \((4-3 i)(2-6 i)\)

Convert the point with the given polar coordinates to rectangular coordinates \((x, y) .\) polar coordinates \(\left(12, \frac{11 \pi}{4}\right)\)

Write each expression in the form \(a+b i,\) where a and b are real numbers. \((8-4 i)(2-3 i)\)

Evaluate \((-3+3 \sqrt{3} i)^{555}\)