91Ó°ÊÓ

Convert the point with the given rectangular coordinates to polar coordinates \((r, \theta) .\) Always choose the angle \(\theta\) to be in the interval \((-\pi, \pi]\). (6,-5)

Write each expression in the form \(a+b i,\) where a and b are real numbers. \(\left(\frac{1}{2}-\frac{\sqrt{3}}{2} i\right)^{3}\)

Describe the subset of the complex plane consisting of the complex numbers \(z\) such that \(z^{3}\) is a real number.

Find coordinates for three different vectors \(\mathbf{u}\), each of which has a direction determined by an angle of \(\frac{\pi}{6}\).

Suppose \(\mathbf{u}\) and \(\mathbf{v}\) are vectors with the same initial point. Explain why \(|\mathbf{u}-\mathbf{v}|\) equals the distance between the endpoint of \(\mathbf{u}\) and the endpoint of \(\mathbf{v}\).

Convert the point with the given rectangular coordinates to polar coordinates \((r, \theta) .\) Always choose the angle \(\theta\) to be in the interval \((-\pi, \pi]\). (-4,1)

Write each expression in the form \(a+b i,\) where a and b are real numbers. \(i^{8001}\)

Describe the subset of the complex plane consisting of the complex numbers \(z\) such that \(z^{3}\) is a positive number.

Describe the subset of the complex plane consisting of the complex numbers \(z\) such that the real part of \(z^{3}\) is a positive number.

Write each expression in the form \(a+b i,\) where a and b are real numbers. \(i^{1003}\)