Chapter 8

\( 0 \mathbf{a}=\mathbf{0}=m \mathbf{0} \quad\)

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ -2-2 i $$

Exer. 23-30: Find the solutions of the equation. $$ x^{6}-64=0 $$

Exer. 21-28: Given that \(a=\langle 2,-3\rangle, \quad b=\langle 3,4\rangle\), and \(c=\langle-1,5\rangle\), find the number. $$ (\mathbf{a}-\mathbf{b}) \cdot(\mathbf{b}+\mathbf{c}) $$

\((-m) \mathbf{a}=-m \mathbf{a}\)

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ 2 \sqrt{3}+2 i $$

Exer. 23-30: Find the solutions of the equation. $$ x^{6}+64=0 $$

\(-(\mathbf{a}+\mathbf{b})=-\mathbf{a}-\mathbf{b} \quad\)

Distances in a baseball diamond A baseball diamond has four bases (forming a square) that are 90 feet apart; the pitcher's mound is \(60.5\) feet from home plate. Approximate the distance from the pitcher's mound to each of the other three bases.

Sighting a forest fire A forest ranger at an observation point \(A\) sights a fire in the direction \(N 27^{\circ} 10^{\prime} \mathrm{E}\). Another ranger at an observation point \(B, 6.0\) miles due east of \(A\), sights the same fire at \(\mathrm{N} 52^{\circ} 40^{\prime} \mathrm{W}\). Approximate the distance from each of the observation points to the fire.

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ 3-3 \sqrt{3} i $$

Exer. 23-30: Find the solutions of the equation. $$ x^{5}+1=0 $$

Exer. 21-28: Given that \(a=\langle 2,-3\rangle, \quad b=\langle 3,4\rangle\), and \(c=\langle-1,5\rangle\), find the number. $$ \operatorname{comp}_{b} \mathbf{c} $$

A rhombus has sides of length 100 centimeters, and the angle at one of the vertices is \(70^{\circ}\). Approximate the lengths of the diagonals to the nearest tenth of a centimeter.

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ -4-4 i $$

Exer. 23-30: Find the solutions of the equation. $$ x^{3}+8 i=0 $$

If \(\mathbf{v}=\langle a, b\rangle\), prove that the magnitude of \(2 \mathbf{v}\) is twice the magnitude of \(\mathbf{v}\).

Exer. 21-28: Given that \(a=\langle 2,-3\rangle, \quad b=\langle 3,4\rangle\), and \(c=\langle-1,5\rangle\), find the number. $$ \operatorname{comp}_{\mathrm{b}}(\mathbf{a}+\mathbf{c}) $$

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ -10+10 i $$

Exer. 23-30: Find the solutions of the equation. $$ x^{3}-64 i=0 $$

If \(\mathbf{v}=\langle a, b\rangle\) and \(k\) is any real number, prove that the magnitude of \(k \mathbf{v}\) is \(|k|\) times the magnitude of \(\mathbf{v}\).

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ -20 i $$

Exer. 23-30: Find the solutions of the equation. $$ x^{5}-243=0 $$

Exer. 29-32: If c represents a constant force, find the work done if the point of application of c moves along the line segment from \(P\) to \(Q\). $$ \mathbf{c}=3 \mathbf{i}+4 \mathbf{j} ; \quad P(0,0), \quad Q(5,-2) $$

Seismology Seismologists investigate the structure of Earth's interior by analyzing seismic waves caused by earthquakes. If the interior of Earth is assumed to be homogeneous, then these waves will travel in straight lines at a constant velocity \(v\). The figure shows a cross-sectional view of Earth, with the epicenter at \(E\) and an observation station at \(S\). Use the law of cosines to show that the time \(t\) for a wave to travel through Earth's interior from \(E\) to \(S\) is given by $$ t=\frac{2 R}{v} \sin \frac{\theta}{2}, $$ where \(R\) is the radius of Earth and \(\theta\) is the indicated angle with vertex at the center of Earth.

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ -6 i $$

Exer. 23-30: Find the solutions of the equation. $$ x^{4}+81=0 $$

Exer. 29-32: If c represents a constant force, find the work done if the point of application of c moves along the line segment from \(P\) to \(Q\). $$ \mathbf{c}=-10 \mathbf{i}+12 \mathbf{j} ; \quad P(0,0), \quad Q(4,7) $$

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ 12 $$

Use Euler's formula to prove De Moivre's theorem.

Exer. 29-32: If c represents a constant force, find the work done if the point of application of c moves along the line segment from \(P\) to \(Q\). $$ \mathbf{c}=6 \mathbf{i}+4 \mathbf{j} ; \quad P(2,-1), \quad Q(4,3) $$ (Hint: Find a vector \(\mathbf{b}=\left\langle b_{1}, b_{2}\right\rangle\) such that \(\mathbf{b}=\overrightarrow{P Q}\).)

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ 15 $$

Exer. 29-32: If c represents a constant force, find the work done if the point of application of c moves along the line segment from \(P\) to \(Q\). $$ \mathbf{c}=-\mathbf{i}+7 \mathbf{j} ; \quad P(-2,5), \quad Q(6,1) $$

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ -7 $$

A constant force of magnitude 4 has the same direction as \(\mathbf{j}\). Find the work done if its point of application moves from \(P(0,0)\) to \(Q(8,3)\).

Approximate the area of triangle \(A B C\). $$\alpha=60^{\circ}, \quad b=20, \quad c=30$$

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ -5 $$

A constant force of magnitude 10 has the same direction as \(-\mathbf{i}\). Find the work done if its point of application moves from \(P(0,1)\) to \(Q(1,0)\).

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ 6 i $$

Exer. \(35-40:\) Prove the property if a and \(b\) are vectors and \(m\) is a real number. $$ \mathbf{a} \cdot \mathbf{a}=\|\mathbf{a}\|^{2} $$

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ 4 i $$

Exer. \(35-40:\) Prove the property if a and \(b\) are vectors and \(m\) is a real number. $$ \mathbf{a} \cdot \mathbf{b}=\mathbf{b} \cdot \mathbf{a} $$

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ -5-5 \sqrt{3} i $$

Exer. \(35-40:\) Prove the property if a and \(b\) are vectors and \(m\) is a real number. $$ (m \mathbf{a}) \cdot \mathbf{b}=m(\mathbf{a} \cdot \mathbf{b}) $$

Approximate the area of triangle \(A B C\). $$\alpha=80.1^{\circ}, \quad a=8.0, \quad b=3.4$$

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ \sqrt{3}-i $$

Exer. \(35-40:\) Prove the property if a and \(b\) are vectors and \(m\) is a real number. $$ m(\mathbf{a} \cdot \mathbf{b})=\mathbf{a} \cdot(m \mathbf{b}) $$

Approximate the area of triangle \(A B C\). $$y=32.1^{\circ}, \quad a=14.6, \quad c=15.8$$

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ 2+i $$

Exer. \(35-40:\) Prove the property if a and \(b\) are vectors and \(m\) is a real number. $$ \mathbf{0} \cdot \mathbf{a}=0 $$