91Ó°ÊÓ

Show that if \(\mathbf{u}\) is a nonzero vector, then the vector \(\frac{\mathbf{u}}{\|\mathbf{u}\|}\) has magnitude \(1 .\)

Show that the formula Area \((A B C)=\frac{1}{2} a b \sin C\) holds if \(A B C\) is a right triangle.

If \(u\) is a nonzero vector, for what values of \(k\) does the equation \(\|k \mathbf{u}\|=k\|\mathbf{u}\|\) hold? Explain.

Convert each of the given polar equations to rectangular form. $$r=2 \cos \theta$$

Convert each of the given polar equations to rectangular form. $$r=4 \sin \theta$$

Convert each of the given polar equations to rectangular form. $$r^{2} \cos 2 \theta=4$$

Convert each of the given polar equations to rectangular form. $$r^{2} \cos 2 \theta=1$$

A patented device converts a radar signal given in polar coordinates to a format in rectangular coordinates so that it is better suited to display in a televisiontype display device. If a radar signal is at the point \(\left(3,-\frac{2 \pi}{3}\right)\) find the exact values of the corresponding rectangular coordinates in the television display.

A boat departs its starting point and travels 4 miles north and 3 miles west. Determine its current location in polar coordinates, using the starting point as the origin. Use a scientific calculator to approximate \(\theta,\) in radians, to three decimal places.

Explain why \((r, \theta)\) and \((r, \theta+2 \pi)\) represent the same point in the polar coordinate system.