Chapter 6

PROOF Prove the following. \(||\mathbf{u - v}||^{2} = ||\mathbf{u}||^{2} + ||\mathbf{v}||^{2} - 2\mathbf{u} \cdot \mathbf{v}\)

In Exercises 83-98, (a) use the formula on page 474 to find the indicated roots of the complex number, (b) represent each of the roots graphically, and (c) write each of the root sin standard form. Fifth roots of \(1\)

PROOF Prove that if \(\mathbf{u}\) is a unit vector and \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{i}\), then \(\mathbf{u} = \cos\ \theta \mathbf{i} + \sin\ \theta \mathbf{j}\).

PROOF Prove that if \(\mathbf{u}\) is a unit vector and \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{j},\) then $$ \mathbf{u}=\cos \left(\frac{\pi}{2}-\theta\right) \mathbf{i}+\sin \left(\frac{\pi}{2}-\theta\right) \mathbf{j} $$

ROPE TENSION To carry a 100 -pound cylindrical weight, two people lift on the ends of short ropes that are tied to an eyelet on the top center of the cylinder. Each rope makes a \(20^{\circ}\) angle with the vertical. Draw a figure that gives a visual representation of the situation, and find the tension in the ropes.

In Exercises 99-106, use the formula on page 474 to find all the solutions of the equation and represent the solutions graphically. \(x^3 + 1 = 0\)

In Exercises 99-106, use the formula on page 474 to find all the solutions of the equation and represent the solutions graphically. \(x^5 + 243 = 0\)

NAVIGATION A commercial jet is flying from Miami to Seattle. The jet's velocity with respect to the air is 580 miles per hour, and its bearing is \(332^{\circ}\). The wind, at the altitude of the plane, is blowing from the southwest with a velocity of 60 miles per hour. (a) Draw a figure that gives a visual representation of the situation. (b) Write the velocity of the wind as a vector in component form. (c) Write the velocity of the jet relative to the air in component form. (d) What is the speed of the jet with respect to the ground? (e) What is the true direction of the jet?

In Exercises 99-106, use the formula on page 474 to find all the solutions of the equation and represent the solutions graphically. \(x^3 - (1 - i) = 0\)

TRUE OR FALSE? In Exercises 107 and 108, determine whether the statement is true or false. Justify your answer. Geometrically, the \(n\)th roots of any complex number \(z\) are all equally spaced around the unit circle centered at the origin.

TRUE OR FALSE? In Exercises 107 and 108, determine whether the statement is true or false. Justify your answer. The product of two complex numbers is zero only when the modulus of one (or both) of the complex numbers is zero.

Given two complex numbers \(z_1 = r_1(\cos\ \theta_1 + i\ \sin\ \theta_1)\) and \(z_2 = r_2(\cos\ \theta_2 + i\ \sin\ \theta_2)\), \(z_2 \neq 0\), show that \(\frac{z_1}{z_2} = \frac{r_1}{r_2}[\cos(\theta_1 - \theta_2) +\ i\ \sin(\theta_1 - \theta_2)]\).

Show that \(\overline{z} = r[\cos(-\theta) + i\ \sin(-\theta)]\) is the complex conjugate of \(z = r(\cos\ \theta + i\ \sin\ \theta)\).

PROOF Prove that \((\cos\ \theta)\mathbf{i} + (\sin\ \theta)\mathbf{j}\) is a unit vector for any value of \(\theta\).

Show that the negative of \(z = r(\cos\ \theta + i\ \sin\ \theta)\) is \(-z = r[\cos(\theta+\pi) + i\ \sin(\theta+\pi)]\).

CAPSTONE The initial and terminal points of vector are \((3, -4)\) and \((9, 1)\), respectively. (a) Write \(\mathbf{v}\) in component form. (b) Write \(\mathbf{v}\) as the linear combination of the standard unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). (c) Sketch \(\mathbf{v}\) with its initial point at the origin. (d) Find the magnitude of \(\mathbf{v}\).

Show that \(\frac{1}{2}(1 - \sqrt{3}i)\) is a ninth root of \(-1\).

GRAPHICAL REASONING Consider two forces \(\mathbf{F}_1 = \langle 10, 0 \rangle\) and \(\mathbf{F}_2 = 5\langle \cos\ \theta, \sin\ \theta \rangle\). (a) Find ||\(\mathbf{F}_1 + \mathbf{F}_2\)|| as a function of \(\theta\). (b) Use a graphing utility to graph the function in part (a) for \(0 \leq \theta < 2\pi\). (c) Use the graph in part (b) to determine the range of the function. What is its maximum, and for what value of \(\theta\) does it occur? What is its minimum, and for what value of \(\theta\) does it occur? (d) Explain why the magnitude of the resultant is never 0.

Show that \(2^{-1/4}(1 - i)\) is a ninth root of \(-2\).

TECHNOLOGY Write a program for your graphing utility that graphs two vectors and their difference given the vectors in component form.

THINK ABOUT IT Explain how you can use DeMoivre's Theorem to solve the polynomial equation \(x^4 + 16 = 0\). [Hint: Write \(-16\) as \(16(\cos\ \pi + i\ \sin\ \pi)\).]

WRITING In your own words, state the difference between a scalar and a vector. Give examples of each.

WRITING Give geometric descriptions of the operations of addition of vectors and multiplication of a vector by a scalar.

WRITING Identify the quantity as a scalar or as a vector. Explain your reasoning. (a) The muzzle velocity of a bullet (b) The price of a company's stock (c) The air temperature in a room (d) The weight of an automobile