Chapter 10

The vector \(\mathbf{v}\) has initial point \(P\) and terminal point \(Q .\) Find its position vector. That is, express \(\mathbf{v}\) in the form \(a \mathbf{i}+b \mathbf{j} .\) $$ P=(1,1) ; \quad Q=(2,2) $$

Plot each point given in polar coordinates. $$ \left(-3,-\frac{\pi}{2}\right) $$

Ramp Angle Billy and Timmy are using a ramp to load furniture into a truck. While rolling a 250 -pound piano up the ramp, they discover that the truck is too full of other furniture for the piano to fit. Timmy holds the piano in place on the ramp while Billy repositions other items to make room for it in the truck. If the angle of inclination of the ramp is \(20^{\circ}\), how many pounds of force must Timmy exert to hold the piano in position?

Write each complex number in rectangular form. $$ 2 e^{i \frac{\pi}{18}} $$

Write each complex number in rectangular form. $$ 3 e^{i \frac{\pi}{10}} $$

Find the acute angle that a constant unit force vector makes with the positive \(x\) -axis if the work done by the force in moving a particle from (0,0) to (4,0) equals 2 .

In Problems \(37-44,\) find \(z w\) and \(\frac{z}{w} .\) Write each answer in polar form and in exponential form. \(z=2\left(\cos \frac{2 \pi}{9}+i \sin \frac{2 \pi}{9}\right)\) \(w=4\left(\cos \frac{\pi}{9}+i \sin \frac{\pi}{9}\right)\)

Prove the distributive property: $$ \mathbf{u} \cdot(\mathbf{v}+\mathbf{w})=\mathbf{u} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{w} $$

Find \(z w\) and \(\frac{z}{w} .\) Write each answer in polar form and in exponential form. \(z=\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\) \(w=\cos \frac{5 \pi}{9}+i \sin \frac{5 \pi}{9}\)

Find \(z w\) and \(\frac{z}{w} .\) Write each answer in polar form and in exponential form. \(z=3 e^{i \frac{13 \pi}{18}}\) \(w=4 e^{i \frac{3 \pi}{2}}\)

Plot each point given in polar coordinates, and find other polar coordinates \((r, \theta)\) of the point for which: (a) \(r>0, \quad-2 \pi \leq \theta<0\) (b) \(r<0, \quad 0 \leq \theta<2 \pi\) (c) \(r>0, \quad 2 \pi \leq \theta<4 \pi\) $$ \left(1, \frac{\pi}{2}\right) $$

Identify and graph each polar equation. $$ r=2+2 \cos \theta $$

Find \(z w\) and \(\frac{z}{w} .\) Write each answer in polar form and in exponential form. \(z=2 e^{i \frac{4 \pi}{9}}\) \(w=6 e^{i \frac{10 \pi}{9}}\)

Plot each point given in polar coordinates, and find other polar coordinates \((r, \theta)\) of the point for which: (a) \(r>0, \quad-2 \pi \leq \theta<0\) (b) \(r<0, \quad 0 \leq \theta<2 \pi\) (c) \(r>0, \quad 2 \pi \leq \theta<4 \pi\) $$ (2, \pi) $$

Identify and graph each polar equation. $$ r=1+\sin \theta $$

Suppose that \(\mathbf{v}\) and \(\mathbf{w}\) are unit vectors. If the angle between \(\mathbf{v}\) and \(\mathbf{i}\) is \(\alpha\) and the angle between \(\mathbf{w}\) and \(\mathbf{i}\) is \(\beta\), use the idea of the dot product \(\mathbf{v} \cdot \mathbf{w}\) to prove that $$ \cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta $$

Plot each point given in polar coordinates, and find other polar coordinates \((r, \theta)\) of the point for which: (a) \(r>0, \quad-2 \pi \leq \theta<0\) (b) \(r<0, \quad 0 \leq \theta<2 \pi\) (c) \(r>0, \quad 2 \pi \leq \theta<4 \pi\). $$ \left(-3,-\frac{\pi}{4}\right) $$

Identify and graph each polar equation. $$ r=3-3 \sin \theta $$

Plot each point given in polar coordinates, and find other polar coordinates \((r, \theta)\) of the point for which: (a) \(r>0, \quad-2 \pi \leq \theta<0\) (b) \(r<0, \quad 0 \leq \theta<2 \pi\) (c) \(r>0, \quad 2 \pi \leq \theta<4 \pi\) $$ \left(-2,-\frac{2 \pi}{3}\right) $$

Let \(\mathbf{v}\) and \(\mathbf{w}\) denote two nonzero vectors. Show that the vectors \(\|\mathbf{w}\| \mathbf{v}+\|\mathbf{v}\| \mathbf{w}\) and \(\|\mathbf{w}\| \mathbf{v}-\|\mathbf{v}\| \mathbf{w}\) are orthogonal.

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$ \left(3, \frac{\pi}{2}\right) $$

Find each quantity if \(\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}\) and \(\mathbf{w}=-2 \mathbf{i}+3 \mathbf{j}\) \(2 v+3 w\)

Identify and graph each polar equation. $$ r=2+\sin \theta $$

Let \(\mathbf{v}\) and \(\mathbf{w}\) denote two nonzero vectors. Show that the vector \(\mathbf{v}-\alpha \mathbf{w}\) is orthogonal to \(\mathbf{w}\) if \(\alpha=\frac{\mathbf{v} \cdot \mathbf{w}}{\|\mathbf{w}\|^{2}}\)

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$ \left(4, \frac{3 \pi}{2}\right) $$

Find each quantity if \(\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}\) and \(\mathbf{w}=-2 \mathbf{i}+3 \mathbf{j}\) \(3 \mathbf{v}-2 \mathbf{w}\)

Find \(z w\) and \(\frac{z}{w} .\) Write each answer in polar form and in exponential form. \(z=1-i\) \(w=1-\sqrt{3} i\)

Identify and graph each polar equation. $$ r=2-\cos \theta $$

Given vectors \(\mathbf{u}=\mathbf{i}+5 \mathbf{j}\) and \(\mathbf{v}=4 \mathbf{i}+y \mathbf{j},\) find \(y\) so that the angle between the vectors is \(60^{\circ} .\)

Find each quantity if \(\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}\) and \(\mathbf{w}=-2 \mathbf{i}+3 \mathbf{j}\) \(\|\mathbf{v}-\mathbf{w}\|\)

In Problems \(45-56,\) write each expression in rectangular form \(x+\) yi and in exponential form \(r e^{i \theta} .\) $$ \left[4\left(\cos \frac{2 \pi}{9}+i \sin \frac{2 \pi}{9}\right)\right]^{3} $$

Identify and graph each polar equation. $$ r=4-2 \cos \theta $$

Given vectors \(\mathbf{u}=x \mathbf{i}+2 \mathbf{j}\) and \(\mathbf{v}=7 \mathbf{i}-3 \mathbf{j}\). find \(x\) so that the angle between the vectors is \(30^{\circ} .\)

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$ (-3, \pi) $$

Find each quantity if \(\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}\) and \(\mathbf{w}=-2 \mathbf{i}+3 \mathbf{j}\) \(\|\mathbf{v}+\mathbf{w}\|\)

Write each expression in rectangular form \(x+\) yi and in exponential form \(r e^{i \theta} .\) $$ \left[3\left(\cos \frac{4 \pi}{9}+i \sin \frac{4 \pi}{9}\right)\right]^{3} $$

Identify and graph each polar equation. $$ r=4+2 \sin \theta $$

Given vectors \(\mathbf{u}=2 x \mathbf{i}+3 \mathbf{j}\) and \(\mathbf{v}=x \mathbf{i}-8 \mathbf{j},\) find \(x\) so that \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal.

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$ \left(6, \frac{5 \pi}{6}\right) $$

Find each quantity if \(\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}\) and \(\mathbf{w}=-2 \mathbf{i}+3 \mathbf{j}\) \(\|\mathbf{v}\|-\|\mathbf{w}\|\)

Write each expression in rectangular form \(x+\) yi and in exponential form \(r e^{i \theta} .\) $$ \left[4\left(\cos \frac{2 \pi}{9}+i \sin \frac{2 \pi}{9}\right)\right]^{3} $$

Identify and graph each polar equation. $$ r=1+2 \sin \theta $$

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$ \left(5, \frac{5 \pi}{3}\right) $$

Find each quantity if \(\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}\) and \(\mathbf{w}=-2 \mathbf{i}+3 \mathbf{j}\) \(\|\mathbf{v}\|+\|\mathbf{w}\|\)

Write each expression in rectangular form \(x+\) yi and in exponential form \(r e^{i \theta} .\) $$ \left[\sqrt{2}\left(\cos \frac{5 \pi}{16}+i \sin \frac{5 \pi}{16}\right)\right]^{4} $$

Identify and graph each polar equation. $$ r=1-2 \sin \theta $$

Find the unit vector in the same direction as \(\mathbf{V}\). \(\mathbf{v}=5 \mathbf{i}\)

Write each expression in rectangular form \(x+\) yi and in exponential form \(r e^{i \theta} .\) $$ \left[\sqrt{3}\left(\cos \frac{\pi}{18}+i \sin \frac{\pi}{18}\right)\right]^{6} $$

Challenge Problem Prove the polarization identity, $$ \|\mathbf{u}+\mathbf{v}\|^{2}-\|\mathbf{u}-\mathbf{v}\|^{2}=4(\mathbf{u} \cdot \mathbf{v}) $$

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$ \left(-2, \frac{2 \pi}{3}\right) $$