Chapter 7

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. $$\mathbf{w}=\langle 3,5\rangle$$

Multiply or divide as indicated, and leave the answer in trigonometric form. $$\frac{4}{3}\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right) \cdot 2\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)$$

Determine whether the given pairs of vectors are orthogonal. $$\mathbf{v}=\langle 3,1\rangle, \mathbf{w}=\langle 0,1\rangle$$

If at least one triangle with the given characteristics exists, solve the triangle; otherwise, state that there is no solution. If there are treo solutions, give both of them. The standard notation for labeling of triangles is used. $$A=50^{\circ}, a=10, b=8$$

Convert each of the given pairs of polar coordinates to a pair of rectangular coordinates. $$\left(-3, \frac{5 \pi}{6}\right)$$

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. $$\mathbf{v}=\langle 1,1.5\rangle$$

Multiply or divide as indicated, and leave the answer in trigonometric form. $$\frac{5\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)}{2\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right)}$$

Determine whether the given pairs of vectors are orthogonal. $$\mathbf{v}=\langle 2,-3\rangle, \mathbf{w}=\langle-6,4\rangle$$

If at least one triangle with the given characteristics exists, solve the triangle; otherwise, state that there is no solution. If there are treo solutions, give both of them. The standard notation for labeling of triangles is used. $$B=70^{\circ}, b=10, c=25$$

Convert each of the given pairs of rectangular coordinates to a pair of polar coordinates ( \(r, \theta\) ) with \(r>0\) and \(0 \leq \theta<2 \pi\). $$(1, \sqrt{3})$$

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. $$\mathbf{u}=\langle-1.5,3\rangle$$

Multiply or divide as indicated, and leave the answer in trigonometric form. $$\frac{6\left(\cos \frac{\pi}{12}+i \sin \frac{\pi}{12}\right)}{3\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)}$$

Determine whether the given pairs of vectors are orthogonal. $$\mathbf{v}=\langle-5,2\rangle, \mathbf{w}=\langle 4,-10\rangle$$

If at least one triangle with the given characteristics exists, solve the triangle; otherwise, state that there is no solution. If there are treo solutions, give both of them. The standard notation for labeling of triangles is used. $$C=48^{\circ}, c=7, b=12$$

Convert each of the given pairs of rectangular coordinates to a pair of polar coordinates ( \(r, \theta\) ) with \(r>0\) and \(0 \leq \theta<2 \pi\). $$(-4,-4)$$

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. $$\mathbf{v}=\left\langle\frac{4}{3}, \frac{2}{5}\right\rangle$$

Use De Moivre's Theorem to find each expression. $$(1+i)^{4}$$

Determine whether the given pairs of vectors are orthogonal. $$\mathbf{v}=\left\langle\frac{1}{3}, 2\right\rangle, \mathbf{w}=\left\langle 6, \frac{5}{2}\right\rangle$$

Convert each of the given pairs of rectangular coordinates to a pair of polar coordinates ( \(r, \theta\) ) with \(r>0\) and \(0 \leq \theta<2 \pi\). $$(-\sqrt{3},-1)$$

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. $$\mathbf{w}=\left\langle-\frac{1}{2},-\frac{1}{4}\right\rangle$$

Use De Moivre's Theorem to find each expression. $$(2-2 i)^{4}$$

Determine whether the given pairs of vectors are orthogonal. $$\mathbf{v}=\langle 3,5\rangle, \mathbf{w}=\left\langle\frac{5}{6}, \frac{1}{2}\right\rangle$$

Convert each of the given pairs of rectangular coordinates to a pair of polar coordinates ( \(r, \theta\) ) with \(r>0\) and \(0 \leq \theta<2 \pi\). $$(-2,2 \sqrt{3})$$

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. $$\mathbf{v}=4 \mathbf{i}-2 \mathbf{j}$$

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

Determine whether the given pairs of vectors are orthogonal. $$\mathbf{v}=\langle 1,0\rangle, \mathbf{w}=\langle 0,3\rangle$$

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

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

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. $$\mathbf{v}=-3 \mathbf{i}+4 \mathbf{j}$$

Use De Moivre's Theorem to find each expression. $$(1-i \sqrt{3})^{6}$$

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

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

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

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. $$\mathbf{v}=1 \mathbf{i}+2.5 \mathbf{j}$$

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

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

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

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 \(r<0\). $$(2, \pi)$$

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. $$\mathbf{v}=-3.2 \mathbf{i}+2 \mathbf{j}$$

Use De Moivre's Theorem to find each expression. $$(-1-i)^{8}$$

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. $$r=1+\sin \theta$$

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 \(r<0\). $$\left(-3, \frac{\pi}{2}\right)$$

Find a unit vector in the same direction as the given vector. $$\mathbf{u}=\langle 3,4\rangle$$

Find the square roots of each complex number. Round all numbers to three decimal places. $$i$$

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

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

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 \(r<0\). $$\left(4, \frac{3 \pi}{2}\right)$$

Find a unit vector in the same direction as the given vector. $$\mathbf{v}=\langle-12,5\rangle$$

Find the square roots of each complex number. Round all numbers to three decimal places. $$-2 i$$