Chapter 10

Write the vector \(\mathbf{v}\) in the form \(\mathbf{ai}+ \mathbf{bj}\), given its magnitude \(\|\mathbf{v}\|\) and the angle \(\alpha\) it makes with the positive \(x\) -axis. \(\|\mathbf{v}\|=3, \quad \alpha=240^{\circ}\)

Find all the complex roots. Write your answers in exponential form. The complex fifth roots of \(-i\)

The rectangular coordinates of a point are given. Find polar coordinates for each point. $$ (5,5 \sqrt{3}) $$

Write the vector \(\mathbf{v}\) in the form \(\mathbf{ai}+ \mathbf{bj}\), given its magnitude \(\|\mathbf{v}\|\) and the angle \(\alpha\) it makes with the positive \(x\) -axis. \(\|\mathbf{v}\|=25, \quad \alpha=330^{\circ}\)

Find the four complex fourth roots of unity, \(1,\) and plot them.

The rectangular coordinates of a point are given. Find polar coordinates for each point. $$ \left(-\frac{\sqrt{3}}{2},-\frac{1}{2}\right) $$

Write the vector \(\mathbf{v}\) in the form \(\mathbf{ai}+ \mathbf{bj}\), given its magnitude \(\|\mathbf{v}\|\) and the angle \(\alpha\) it makes with the positive \(x\) -axis. \(\|\mathbf{v}\|=15, \quad \alpha=315^{\circ}\)

The rectangular coordinates of a point are given. Find polar coordinates for each point. $$ (1.3,-2.1) $$

Find the direction angle of \(\mathbf{v}\). \(\mathbf{v}=3 \mathbf{i}+3 \mathbf{j}\)

Show that each complex \(n\) th root of a nonzero complex number \(w\) has the same magnitude.

Graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph. $$ r=8 \cos \theta ; r=2 \sec \theta $$

Find the direction angle of \(\mathbf{v}\). \(\mathbf{v}=\mathbf{i}+\sqrt{3} \mathbf{j}\)

The rectangular coordinates of a point are given. Find polar coordinates for each point. $$ (-0.8,-2.1) $$

Graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph. $$ r=8 \sin \theta ; r=4 \csc \theta $$

Find the direction angle of \(\mathbf{v}\). \(\mathbf{v}=-3 \sqrt{3} \mathbf{i}+3 \mathbf{j}\)

The rectangular coordinates of a point are given. Find polar coordinates for each point. $$ (8.3,4.2) $$

Graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph. $$ r=\sin \theta ; r=1+\cos \theta $$

Find the direction angle of \(\mathbf{v}\). \(\mathbf{v}=-5 \mathbf{i}-5 \mathbf{j}\)

Graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph. $$ r=3 ; r=2+2 \cos \theta $$

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ 2 x^{2}+2 y^{2}=3 $$

Find the direction angle of \(\mathbf{v}\). \(\mathbf{v}=4 \mathbf{i}-2 \mathbf{j}\)

Prove \(r e^{i \theta}=r e^{i(\theta+2 k \pi)}, k\) an integer.

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ x^{2}+y^{2}=x $$

Find the direction angle of \(\mathbf{v}\). \(\mathbf{v}=6 \mathbf{i}-4 \mathbf{j}\)

Show that \(e^{i \pi}+1=0\)

Graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph. $$ r=1+\cos \theta ; r=3 \cos \theta $$

Find the direction angle of \(\mathbf{v}\). \(\mathbf{v}=-\mathbf{i}-5 \mathbf{j}\)

Prove that De Moivre's Theorem is true for \(a l l\) integers \(n\) by assuming it is true for integers \(n \geq 1\) and then showing it is true for 0 and for negative integers. nHint: Multiply the numerator and the denominator by the conjugate of the denominator, and use even-odd properties.

Graph each polar equation. $$ r=\frac{2}{1-\cos \theta} \quad(\text {parabola}) $$

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ y^{2}=2 x $$

Find the direction angle of \(\mathbf{v}\). \(\mathbf{v}=-\mathbf{i}+3 \mathbf{j}\)

(a) Consider the expression \(a_{n}=\left(a_{n-1}\right)^{2}+z,\) where \(z\) is some complex number (called the seed) and \(a_{0}=z\). Compute \(a_{1}\left(=a_{0}^{2}+z\right), a_{2}\left(=a_{1}^{2}+z\right), a_{3}\left(=a_{2}^{2}+z\right), a_{4}, a_{5}\) and \(a_{6}\) for the following seeds: \(z_{1}=0.1-0.4 i\), \(z_{2}=0.5+0.8 i, \quad z_{3}=-0.9+0.7 i, \quad z_{4}=-1.1+0.1 i\) \(z_{5}=0-1.3 i,\) and \(z_{6}=1+1 i\) (b) The dark portion of the graph represents the set of all values \(z=x+y i\) that are in the Mandelbrot set. Determine which complex numbers in part (a) are in this set by plotting them on the graph. Do the complex numbers that are not in the Mandelbrot set have any common characteristics regarding the values of \(a_{6}\) found in part (a)? (c) Compute \(|z|=\sqrt{x^{2}+y^{2}}\) for each of the complex numbers in part (a). Now compute \(\left|a_{6}\right|\) for each of the complex numbers in part (a). For which complex numbers is \(\left|a_{6}\right| \leq|z|\) and \(|z| \leq 2 ?\) Conclude that the criterion for a complex number to be in the Mandelbrot set is that \(\left|a_{n}\right| \leq|z|\) and \(|z| \leq 2\).

A child pulls a wagon with a force of 40 pounds. The handle of the wagon makes an angle of \(30^{\circ}\) with the ground. Express the force vector \(\mathbf{F}\) in terms of i and \(\mathbf{j}\).

Graph each polar equation. $$ r=\frac{1}{3-2 \cos \theta} \quad(\text {ellipse}) $$

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ 4 x^{2} y=1 $$

A man pushes a wheelbarrow up an incline of \(20^{\circ}\) with a force of 100 pounds. Express the force vector \(\mathbf{F}\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\).

Graph each polar equation. $$ r=\frac{1}{1-\cos \theta} \quad(\text {parabola}) $$

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ x=4 $$

Problems \(77-86\) are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the area of the triangle with \(a=8, b=11\), and \(C=113^{\circ}\).

Graph each polar equation. $$ r=\theta, \quad \theta \geq 0 \quad(\text {spiral of Archimedes}) $$

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ y=-3 $$

Graph each polar equation. $$ r=\frac{3}{\theta} \quad(\text {reciprocal spiral}) $$

A Boeing 787 Dreamliner maintains a constant airspeed of 550 miles per hour (mph) headed due north. The jet stream is \(100 \mathrm{mph}\) in the northeasterly direction. (a) Express the velocity \(\mathbf{v}_{\mathrm{a}}\) of the 787 relative to the air and the velocity \(\mathbf{v}_{\mathrm{w}}\) of the jet stream in terms of \(\mathbf{i}\) and \(\mathbf{j}\). (b) Find the velocity of the 787 relative to the ground. (c) Find the actual speed and direction of the 787 relative to the ground.

Graph each polar equation. $$ r=\csc \theta-2, \quad 0<\theta<\pi \quad(\text { conchoid }) $$

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ r=\sin \theta+1 $$

An Airbus A 320 jet maintains a constant airspeed of \(500 \mathrm{mph}\) headed due west. The jet stream is \(100 \mathrm{mph}\) in the southeasterly direction. (a) Express the velocity \(\mathbf{v}_{\text {a }}\) of the A320 relative to the air and the velocity \(\mathbf{v}_{\mathrm{w}}\) of the jet stream in terms of i and \(\mathbf{j}\). (b) Find the velocity of the \(\mathrm{A} 320\) relative to the ground. (c) Find the actual speed and direction of the \(\mathrm{A} 320\) relative to the ground.

Airplane An airplane has an airspeed of 500 kilometers per hour \((\mathrm{km} / \mathrm{h})\) bearing \(\mathrm{N} 45^{\circ} \mathrm{E}\). The wind velocity is \(60 \mathrm{~km} / \mathrm{h}\) in the direction \(\mathrm{N} 30^{\circ} \mathrm{W}\). Find the resultant vector representing the path of the plane relative to the ground. What is the groundspeed of the plane? What is its direction?

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ r^{2}=\cos \theta $$

Graph each polar equation. $$ r=\tan \theta, \quad-\frac{\pi}{2}<\theta<\frac{\pi}{2} \quad(\text {kappa curve}) $$

An airplane has an airspeed of \(600 \mathrm{~km} / \mathrm{h}\) bearing \(\mathrm{S} 30^{\circ} \mathrm{E}\). The wind velocity is \(40 \mathrm{~km} / \mathrm{h}\) in the direction \(\mathrm{S} 45^{\circ} \mathrm{E}\). Find the resultant vector representing the path of the plane relative to the ground. What is the groundspeed of the plane? What is its direction?