Chapter 10

Find \(b\) so that the vectors \(\mathbf{v}=\mathbf{i}+\mathbf{j}\) and \(\mathbf{w}=\mathbf{i}+b \mathbf{j}\) are orthogonal.

Plot each complex number in the complex plane and write it in polar form and in exponential form. $$ 9 \sqrt{3}+9 i $$

Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $$ r \cos \theta=4 $$

Decompose \(\mathbf{v}\) into two vectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\), where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\), and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}\). $$ \mathbf{v}=2 \mathbf{i}-3 \mathbf{j}, \quad \mathbf{w}=\mathbf{i}-\mathbf{j} $$

Plot each complex number in the complex plane and write it in polar form and in exponential form. $$ 3-4 i $$

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

Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $$ r \cos \theta=-2 $$

Decompose \(\mathbf{v}\) into two vectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\), where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\), and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}\). $$ \mathbf{v}=-3 \mathbf{i}+2 \mathbf{j}, \quad \mathbf{w}=2 \mathbf{i}+\mathbf{j} $$

Plot each complex number in the complex plane and write it in polar form and in exponential form. $$ 2+\sqrt{3} i $$

Plot each point given in polar coordinates. $$ \left(4, \frac{3 \pi}{2}\right) $$

Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $$ r \sin \theta=-2 $$

Decompose \(\mathbf{v}\) into two vectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\), where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\), and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}\). $$ \mathbf{v}=\mathbf{i}-\mathbf{j}, \quad \mathbf{w}=-\mathbf{i}-2 \mathbf{j} $$

Decompose \(\mathbf{v}\) into two vectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\), where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\), and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}\). $$ \mathbf{v}=2 \mathbf{i}-\mathbf{j}, \quad \mathbf{w}=\mathbf{i}-2 \mathbf{j} $$

Plot each complex number in the complex plane and write it in polar form and in exponential form. $$ \sqrt{5}-i $$

Plot each point given in polar coordinates. $$ (-3, \pi) $$

Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $$ r=2 \sin \theta $$

Decompose \(\mathbf{v}\) into two vectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\), where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\), and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}\). $$ \mathbf{v}=3 \mathbf{i}+\mathbf{j}, \quad \mathbf{w}=-2 \mathbf{i}-\mathbf{j} $$

In Problems 25-36, write each complex number in rectangular form. $$ 2\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right) $$

Plot each point given in polar coordinates. $$ \left(6, \frac{\pi}{6}\right) $$

Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $$ r=-4 \sin \theta $$

Decompose \(\mathbf{v}\) into two vectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\), where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\), and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}\). $$ \mathbf{v}=\mathbf{i}-3 \mathbf{j}, \quad \mathbf{w}=4 \mathbf{i}-\mathbf{j} $$

Write each complex number in rectangular form. $$ 3\left(\cos \frac{7 \pi}{6}+i \sin \frac{7 \pi}{6}\right) $$

Plot each point given in polar coordinates. $$ \left(5, \frac{5 \pi}{3}\right) $$

Find a vector of magnitude 15 that is parallel to \(4 \mathbf{i}-3 \mathbf{j}\)

The vector \(\mathbf{v}\) has initial point \(P\) and terminal point \(Q .\) Find its position vector. That is, express \(\mathbf{v}\) in the form \(a \mathbf{i}+b \mathbf{j} .\) $$ P=(0,0) ; \quad Q=(3,4) $$

Write each complex number in rectangular form. $$ 4 e^{i \frac{7 \pi}{4}} $$

Find a vector of magnitude 5 that is parallel to \(-12 \mathbf{i}+9 \mathbf{j}\)

The vector \(\mathbf{v}\) has initial point \(P\) and terminal point \(Q .\) Find its position vector. That is, express \(\mathbf{v}\) in the form \(a \mathbf{i}+b \mathbf{j} .\) $$ P=(0,0) ; \quad Q=(-3,-5) $$

Write each complex number in rectangular form. $$ 2 e^{i \frac{5 \pi}{6}} $$

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

Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $$ r \csc \theta=8 $$

Computing Work Find the work done by a force of 3 pounds acting in a direction of \(60^{\circ}\) to the horizontal in moving an object 6 feet from (0,0) to (6,0)

The vector \(\mathbf{v}\) has initial point \(P\) and terminal point \(Q .\) Find its position vector. That is, express \(\mathbf{v}\) in the form \(a \mathbf{i}+b \mathbf{j} .\) $$ P=(3,2) ; \quad Q=(5,6) $$

Write each complex number in rectangular form. $$ 3\left(\cos \frac{3 \pi}{2}+i \sin \frac{3 \pi}{2}\right) $$

Plot each point given in polar coordinates. $$ \left(4,-\frac{2 \pi}{3}\right) $$

Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $$ r \csc \theta=-2 $$

Computing Work A wagon is pulled horizontally by exerting a force of 20 pounds on the handle at an angle of \(30^{\circ}\) with the horizontal. How much work is done in moving the wagon 100 feet?

The vector \(\mathbf{v}\) has initial point \(P\) and terminal point \(Q .\) Find its position vector. That is, express \(\mathbf{v}\) in the form \(a \mathbf{i}+b \mathbf{j} .\) $$ P=(-3,2) ; \quad Q=(6,5) $$

Plot each point given in polar coordinates. $$ \left(2,-\frac{5 \pi}{4}\right) $$

Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation. $$ r \sec \theta=-4 $$

Solar Energy The amount of energy collected by a solar panel depends on the intensity of the sun's rays and the area of the panel. Let the vector I represent the intensity, in watts per square centimeter, having the direction of the sun's rays. Let the vector \(\mathbf{A}\) represent the area, in square centimeters, whose direction is the orientation of a solar panel. See the figure. The total number of watts collected by the panel is given by \(W=|\mathbf{I} \cdot \mathbf{A}|\) Suppose that \(\mathbf{I}=\langle-0.02,-0.01\rangle\) and \(\mathbf{A}=\langle 300,400\rangle\) (a) Find \(\|\mathbf{I}\|\) and \(\|\mathbf{A}\|,\) and interpret the meaning of each. (b) Compute \(W\) and interpret its meaning. (c) If the solar panel is to collect the maximum number of watts, what must be true about I and \(\mathbf{A}\) ?

The vector \(\mathbf{v}\) has initial point \(P\) and terminal point \(Q .\) Find its position vector. That is, express \(\mathbf{v}\) in the form \(a \mathbf{i}+b \mathbf{j} .\) $$ P(-2,-1) ; \quad Q=(6,-2) $$

Write each complex number in rectangular form. $$ 7 e^{i \pi} $$

The vector \(\mathbf{v}\) has initial point \(P\) and terminal point \(Q .\) Find its position vector. That is, express \(\mathbf{v}\) in the form \(a \mathbf{i}+b \mathbf{j} .\) $$ P=(-1,4) ; \quad Q=(6,2) $$

Write each complex number in rectangular form. $$ 3 e^{i \frac{\pi}{2}} $$

Plot each point given in polar coordinates. $$ \left(-3,-\frac{3 \pi}{4}\right) $$

The vector \(\mathbf{v}\) has initial point \(P\) and terminal point \(Q .\) Find its position vector. That is, express \(\mathbf{v}\) in the form \(a \mathbf{i}+b \mathbf{j} .\) $$ P=(1,0) ; \quad Q=(0,1) $$

Write each complex number in rectangular form. $$ 0.2\left(\cos \frac{5 \pi}{9}+i \sin \frac{5 \pi}{9}\right) $$

Plot each point given in polar coordinates. $$ (-2,-\pi) $$

Braking Load A Chevrolet Silverado with a gross weight of 4500 pounds is parked on a street with a \(10^{\circ}\) grade. Find the magnitude of the force required to keep the Silverado from rolling down the hill. What is the magnitude of the force perpendicular to the hill?