Chapter 9

In Exercises \(53-64,\) perform the indicated multiplication or division. Express your answer in both polar form \(r(\cos \theta+i \sin \theta)\) and rectangular form \(a+b i\). $$\frac{\cos \pi+i \sin \pi}{\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}}$$

In Exercises \(53-64,\) perform the indicated multiplication or division. Express your answer in both polar form \(r(\cos \theta+i \sin \theta)\) and rectangular form \(a+b i\). $$\frac{\cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}}{\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}}$$

Deal with an object on an inclined plane. The situation is similar to that in Figure \(9-20\) of Example \(12,\) where \(\|\overline{T P}\|\) is the component of the weight of the object parallel to the plane and \(\|\overline{T Q}\|\) is the component of the weight perpendicular to the plane. An object weighing 50 pounds lies on an inclined plane that makes a \(40^{\circ}\) angle with the horizontal. Find the components of the weight parallel and perpendicular to the plane.

In Exercises \(65-72,\) convert to polar form and then multiply or divide. Express your answer in polar form. $$(1+i)(1+\sqrt{3} i)$$

Find the course and ground speed of the plane under the given conditions. Air speed 400 mph in the direction \(150^{\circ} ;\) wind speed \(30 \mathrm{mph}\) from the direction \(60^{\circ}\).

In Exercises \(65-72,\) convert to polar form and then multiply or divide. Express your answer in polar form. $$(1-i)(3-3 i)$$

In Exercises \(65-72,\) convert to polar form and then multiply or divide. Express your answer in polar form. $$\frac{1+i}{1-i}$$

In Exercises \(65-72,\) convert to polar form and then multiply or divide. Express your answer in polar form. $$\frac{2-2 i}{-1-i}$$

In Exercises \(65-72,\) convert to polar form and then multiply or divide. Express your answer in polar form. $$3 i(2 \sqrt{3}+2 i)$$

A plane is flying in the direction \(200^{\circ}\) with an air speed of \(500 \mathrm{mph}\). Its course and ground speed are \(210^{\circ}\) and \(450 \mathrm{mph}\) respectively. What are the direction and speed of the wind?

In Exercises \(65-72,\) convert to polar form and then multiply or divide. Express your answer in polar form. $$\frac{-4 i}{\sqrt{3}+i}$$

A river flows from east to west. A swimmer on the south bank wants to swim to a point on the opposite shore directly north of her starting point. She can swim at \(2.8 \mathrm{mph},\) and there is a 1 -mph current in the river. In what direction should she head so as to travel directly north (that is, what angle should her path make with the south bank of the river)?

In Exercises \(65-72,\) convert to polar form and then multiply or divide. Express your answer in polar form. $$i(i+1)(-\sqrt{3}+i)$$

A river flows from west to east. A swimmer on the north bank swims at \(3.1 \mathrm{mph}\) along a straight course that makes a \(75^{\circ}\) angle with the north bank of the river and reaches the south bank at a point directly south of his starting point. How fast is the current in the river?

In Exercises \(65-72,\) convert to polar form and then multiply or divide. Express your answer in polar form. $$(1-i)(2 \sqrt{3}-2 i)(-4-4 \sqrt{3} i)$$

Explain what is meant by saying that multiplying a complex number \(z=r(\cos \theta+i \sin \theta)\) by \(i\) amounts to rotating \(z 90^{\circ}\) counterclockwise around the origin. [Hint: Express \(i\) and \(i z\) in polar form. What are their relative positions in the complex plane?]

Describe what happens geometrically when you multiply a complex number by 2.

Do Exercise 74 when the weight is 600 pounds and the angles are \(28^{\circ}\) and \(38^{\circ}\).

The sum of two distinct complex numbers, \(a+b i\) and \(c+d i,\) can be found geometrically by means of the socalled parallelogram rule: Plot the points \(a+b i\) and \(c+d i\) in the complex plane, and form the parallelogram, three of whose vertices are \(0, a+b i,\) and \(c+d i,\) as in the figure. Then the fourth vertex of the parallelogram is the point whose coordinate is the sum $$(a+b i)+(c+d i)=(a+c)+(b+d) i$$ (GRAPH CAN'T COPY). Complete the following proof of the parallelogram rule when \(a \neq 0\) and \(c \neq 0\) (a) Find the slope of the line \(K\) from 0 to \(a+b i .[\text { Hint: } K\) contains the points \((0,0) \text { and }(a, b) .]\) (b) Find the slope of the line \(N\) from 0 to \(c+d i\) (c) Find the equation of the line \(L\) through \(a+b i\) and parallel to line \(N\) of part (b). [Hint: The point \((a, b)\) is on \(L\) find the slope of \(L\) by using part (b) and facts about the slope of parallel lines.] (d) Find the equation of the line \(M\) through \(c+d i\) and parallel to line \(K\) of part (a). (e) Label the lines \(K, L, M,\) and \(N\) in the figure. (f) Show by using substitution that the point \((a+c, b+d)\) satisfies both the equation of line \(L\) and the equation of line \(M .\) Therefore, \((a+c, b+d)\) lies on both \(L\) and \(M\) since the only point on both \(L\) and \(M\) is the fourth vertex of the parallelogram (see the figure), this vertex must be \((a+c, b+d) .\) Hence, this vertex has coordinate $$(a+c)+(b+d) i=(a+b i)+(c+d i)$$

Let \(z=a+b i\) be a complex number and denote its conjugate \(a-b i\) by \(\bar{z} .\) Prove that \(|z|^{2}=z \bar{z}\).

Let \(\mathbf{v}\) be the vector with initial point \(\left(x_{1}, y_{1}\right)\) and terminal point \(\left(x_{2}, y_{2}\right),\) and let \(k\) be any real number. (a) Find the component form of \(\mathbf{v}\) and \(k \mathbf{v}\). (b) Calculate \(\|\mathbf{v}\|\) and \(\|k \mathbf{v}\|\). (c) Use the fact that \(\sqrt{k^{2}}=|k|\) to verify that \(\|k \mathbf{v}\|=|k| \cdot\|\mathbf{v}\|\). (d) Show that \(\tan \theta=\tan \beta,\) where \(\theta\) is the direction angle of \(\mathbf{v}\) and \(\beta\) is the direction angle of \(k \mathbf{v} .\) Use the fact that \(\tan t=\tan \left(t+180^{\circ}\right)\) to conclude that \(\mathbf{v}\) and \(k \mathbf{v}\) have either the same or opposite directions. (e) Use the fact that \((c, d)\) and \((-c,-d)\) lie on the same straight line on opposite sides of the origin (Exercise 85 in Section 1.3 ) to verify that \(\mathbf{v}\) and \(k \mathbf{v}\) have the same direction if \(k>0\) and opposite directions if \(k<0\).

Proof of the polar division rule. Let \(z_{1}=r_{1}\left(\cos \theta_{1}+i \sin \theta_{1}\right)\) and \(z_{2}=r_{2}\left(\cos \theta_{2}+i \sin \theta_{2}\right) .\) Then $$\begin{aligned} \frac{z_{1}}{z_{2}} &=\frac{r_{1}\left(\cos \theta_{1}+i \sin \theta_{1}\right)}{r_{2}\left(\cos \theta_{2}+i \sin \theta_{2}\right)} \\ &=\frac{r_{1}\left(\cos \theta_{1}+i \sin \theta_{1}\right)}{r_{2}\left(\cos \theta_{2}+i \sin \theta_{2}\right)} \cdot \frac{\cos \theta_{2}-i \sin \theta_{2}}{\cos \theta_{2}-i \sin \theta_{2}} \end{aligned}$$ (a) Multiply out the denominator on the right side and use the Pythagorean identity to show that it is just the number \(r_{2}\) (b) Multiply out the numerator on the right side; use the subtraction identities for sine and cosine (page 524 ) to show that it is $$r_{1}\left[\cos \left(\theta_{1}-\theta_{2}\right)+i \sin \left(\theta_{1}-\theta_{2}\right)\right]$$ Therefore, $$\frac{z_{1}}{z_{2}}=\left(\frac{r_{1}}{r_{2}}\right)\left[\cos \left(\theta_{1}-\theta_{2}\right)+i \sin \left(\theta_{1}-\theta_{2}\right)\right]$$

(a) If \(s(\cos \beta+i \sin \beta)=r(\cos \theta+i \sin \theta),(\) with \(r > 0, s > 0\) ), explain why we must have \(s=r\) [Hint: Think distance. \(]\) (b) If \(r(\cos \beta+i \sin \beta)=r(\cos \theta+i \sin \theta),\) explain why \(\cos \beta=\cos \theta\) and \(\sin \beta=\sin \theta .\) [Hint: See property 5 of the complex numbers on page \(322 .]\) (c) If \(\cos \beta=\cos \theta\) and \(\sin \beta=\sin \theta,\) show that angles \(\beta\) and \(\theta\) in standard position have the same terminal side. \([\text {Hint: }(\cos \beta, \sin \beta) \text { and }(\cos \theta, \sin \theta)\) are points on the unit circle. \(]\) (d) Use parts (a)-(c) to prove this equality rule for polar form: $$s(\cos \beta+i \sin \beta)=r(\cos \theta+i \sin \theta)$$ exactly when \(s=r\) and \(\beta=\theta+2 k \pi\) for some integer \(k .\) [Hint: Angles with the same terminal side must differ by an integer multiple of \(2 \pi .]\)