91Ó°ÊÓ

Convert each of the given pairs of rectangular coordinates to a pair of polar coordinates ( \(r, \theta\) ) with \(r>0\) and \(0 \leq \theta<2 \pi\). $$(-1,0)$$

In Exercises \(31-46,\) sketch the graphs of the polar equations. $$r=5 \sin 2 \theta$$

Determine whether the given pairs of vectors are orthogonal. $$\mathbf{v}=\langle 2,0\rangle, \mathbf{w}=\langle 0,4\rangle$$

For each of the points given in polar coordinates, find two additional pairs of polar coordinates \((r, \theta),\) one with \(r>0\) and one with \(r<0\). $$(2, \pi)$$

In Exercises \(31-46,\) sketch the graphs of the polar equations. $$r=1+\cos \theta$$

Find the magnitude and direction of each of the given vectors. Express the direction as an angle \(\theta\) in standard position, where \(0^{\circ} \leq \theta<360^{\circ},\) to tuo decimal places. $$\mathbf{v}=1 \mathbf{i}+2.5 \mathbf{j}$$

Perform each operation, given \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) $$\mathbf{u}=\langle 3,2\rangle, \mathbf{v}=\langle-1,4\rangle, \mathbf{w}=\langle-2,-1\rangle$$ $$2 u+3 w$$

Use De Moivre's Theorem to find each expression. $$(-3-3 i \sqrt{3})^{3}$$

Perform each operation, given \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) $$\mathbf{u}=\langle 3,2\rangle, \mathbf{v}=\langle-1,4\rangle, \mathbf{w}=\langle-2,-1\rangle$$ $$-4 u+v$$

In Exercises \(31-46,\) sketch the graphs of the polar equations. $$r=1+\sin \theta$$