91Ó°ÊÓ

Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$3\left(\cos \frac{3 \pi}{2}+i \sin \frac{3 \pi}{2}\right)$$

Find a vector that has the same direction as \(\langle- 8,2\rangle\) and (a) twice the magnitude (b) one-half the magnitude

Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$\sqrt{34} \text { cis }\left(\tan ^{-1} \frac{3}{5}\right)$$

Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$\sqrt{53} \operatorname{cis}\left[\tan ^{-1}\left(-\frac{2}{7}\right)\right]$$

Find a vector of magnitude 6 that has the opposite direction of \(\mathbf{a}=4 \mathbf{i}-7 \mathbf{j}\)

Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$\sqrt{5} \operatorname{cis}\left[\tan ^{-1}\left(-\frac{1}{2}\right)\right]$$

Find a vector of magnitude 5 that has the opposite direction of \(\mathbf{a}=\langle- 3,-1\rangle\)

Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$\sqrt{10} \text { cis }\left(\tan ^{-1} 3\right)$$

Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$\sqrt{65} \mathrm{cis}\left[\tan ^{-1}\left(-\frac{7}{4}\right)+\pi\right]$$

If forces \(F_{1}, F_{2}, \ldots, F_{n}\) act at a point \(P,\) the net (or resultant) force \(\mathbf{F}\) is the \(\operatorname{sum} \mathbf{F}_{1}+\mathbf{F}_{2}+\cdots+\mathbf{F}_{n} .\) If \(\mathbf{F}=\mathbf{0},\) the forces are said to be in equilibrium. The given forces act at the origin \(O\) of an \(x y\) -plane. (a) Find the net force \(\mathbf{F}\). (b) Find an additional force \(G\) such that equilibrium occurs. $$\mathbf{F}_{1}=\langle 4,3\rangle, \quad \mathbf{F}_{2}=\langle- 2,-3\rangle, \quad \mathbf{F}_{3}=\langle 5,2\rangle$$