/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 62 \(57-64=\) Write \(z_{1}\) and \... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 62

\(57-64=\) Write \(z_{1}\) and \(z_{2}\) in polar form, and then find the product \(z_{1} z_{2}\) and the quotients \(z_{1} / z_{2}\) and 1\(/ z_{1}\) . $$ z_{1}=4 \sqrt{3}-4 i, \quad z_{2}=8 i $$

Short Answer

Expert verified
Product: 32 + 32i鈭�3, Quotient: -1/2 - i鈭�3/2, Reciprocal: 鈭�3/16 + i/16.

Step by step solution

01

Convert to polar form

First, we find the polar form of \( z_1 = 4\sqrt{3} - 4i \). Calculate its magnitude: \( r_1 = \sqrt{(4\sqrt{3})^2 + (-4)^2} = \sqrt{48 + 16} = \sqrt{64} = 8 \). The argument \( \theta_1 \) is given by \( \tan^{-1} \left(\frac{-4}{4\sqrt{3}}\right) = \tan^{-1} \left(-\frac{1}{\sqrt{3}}\right) = -\frac{\pi}{6} \). So, \( z_1 = 8(\cos(-\frac{\pi}{6}) + i\sin(-\frac{\pi}{6})) \).
02

Convert second complex number to polar

For \( z_2 = 8i \), its magnitude is \( r_2 = \sqrt{0^2 + 8^2} = \sqrt{64} = 8 \), and its argument is \( \theta_2 = \frac{\pi}{2} \) since it lies on the positive imaginary axis. So, \( z_2 = 8(\cos(\frac{\pi}{2}) + i\sin(\frac{\pi}{2})) \).
03

Find the product \( z_1 z_2 \)

The product of two complex numbers in polar form is found by multiplying their magnitudes and adding their arguments. Therefore, \( z_1z_2 = (8 \times 8)[\cos(-\frac{\pi}{6} + \frac{\pi}{2}) + i\sin(-\frac{\pi}{6} + \frac{\pi}{2})] = 64[\cos(\frac{\pi}{3}) + i\sin(\frac{\pi}{3})] = 64[\frac{1}{2} + i\frac{\sqrt{3}}{2}] = 32 + 32i\sqrt{3} \).
04

Find the quotient \( \frac{z_1}{z_2} \)

To find the quotient, divide the magnitudes and subtract the arguments: \( \frac{z_1}{z_2} = \frac{8}{8}[\cos(-\frac{\pi}{6} - \frac{\pi}{2}) + i\sin(-\frac{\pi}{6} - \frac{\pi}{2})] = 1[\cos(-\frac{2\pi}{3}) + i\sin(-\frac{2\pi}{3})] = -\frac{1}{2} - i\frac{\sqrt{3}}{2} \).
05

Find \( \frac{1}{z_1} \)

To find \( \frac{1}{z_1} \), take the reciprocal of the magnitude and change the sign of the argument: \( \frac{1}{z_1} = \frac{1}{8}[\cos(\frac{\pi}{6}) + i\sin(\frac{\pi}{6})] = \frac{1}{8}[\frac{\sqrt{3}}{2} + i\frac{1}{2}] = \frac{\sqrt{3}}{16} + \frac{i}{16} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Form
Complex numbers can be expressed in rectangular form (as used in math class), or in polar form, which is useful for multiplying and dividing them.
In polar form, a complex number is represented as its magnitude (distance from the origin) and angle (direction from the positive x-axis) in the complex plane.
A complex number in polar form is written as:
  • \( z = r(\cos(\theta) + i\sin(\theta)) \)
  • "\(r\)" is the magnitude.
  • "\(\theta\)" is the argument (angle).
Using Euler鈥檚 formula, it can be simplified to: \( z = re^{i\theta} \).
It makes operations like multiplication and division simpler, as we only need to work with the magnitudes and angles.
Magnitude
The magnitude, or modulus, of a complex number tells us how far the point is from the origin in the complex plane.
For a complex number \( z = a + bi \), its magnitude is:
\[ r = \sqrt{a^2 + b^2} \]
If you imagine the complex plane as a graph, the magnitude is the length of the line from the origin (0,0) to the point \((a,b)\).
For the given exercise, we calculated the magnitudes as:
  • \( r_1 = 8 \) for \( z_1 = 4\sqrt{3} - 4i \)
  • \( r_2 = 8 \) for \( z_2 = 8i \)
This step is crucial before moving to polar form.
Argument
The argument of a complex number refers to the angle it makes with the positive x-axis in the complex plane.
This angle is usually measured in radians, ranging from \(0\) to \(2\pi\) for a full circle.
For a complex number \( z = a + bi \), the argument \( \theta \) is calculated using:
\[ \theta = \tan^{-1} \left( \frac{b}{a} \right) \]
Adjustments might be needed based on the quadrant in which the complex number lies.
In our example, the arguments calculated are:
  • \( \theta_1 = -\frac{\pi}{6} \) for \( z_1 = 4\sqrt{3} - 4i \)
  • \( \theta_2 = \frac{\pi}{2} \) for \( z_2 = 8i \)
Understanding the argument helps pinpoint the direction of the number.
Complex Multiplication
Complex multiplication is simplified by expressing numbers in polar form.
When multiplying two complex numbers in polar form, you:
  • Multiply their magnitudes: \( r_1r_2 \)
  • Add their arguments: \( \theta_1 + \theta_2 \)
This is due to the properties of sine and cosine during multiplication, making it more intuitive and streamlined.
In the example, for multiplying \( z_1 \) and \( z_2 \), the product is expressed as:\[ z_1z_2 = 64 \left( \cos(\frac{\pi}{3}) + i\sin(\frac{\pi}{3}) \right) \]
Resulting in the rectangular form of \( 32 + 32i\sqrt{3} \). Understanding this helps apply the concept of using polar coordinates efficiently in complex operations.

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Most popular questions from this chapter

The Distance Formula in Polar Coordinates (a) Use the Law of Cosines to prove that the distance between the polar points \(\left(r_{1}, \theta_{1}\right)\) and \(\left(r_{2}, \theta_{2}\right)\) is $$ d=\sqrt{r_{1}^{2}+r_{2}^{2}-2 r_{1} r_{2} \cos \left(\theta_{2}-\theta_{1}\right)} $$ (b) Find the distance between the points whose polar coordinates are \((3,3 \pi / 4)\) and \((1,7 \pi / 6),\) using the formula from part (a). (c) Now convert the points in part (b) to rectangular coordinates. Find the distance between them using the usual Distance Formula. Do you get the same answer?

True Velocity of a Jet A jet is flying through a wind that is blowing with a speed of 55 \(\mathrm{mi} / \mathrm{h}\) in the direction \(\mathrm{N} 30^{\circ} \mathrm{E}\) (see the figure). The jet has a speed of 765 \(\mathrm{mi} / \mathrm{h}\) relative to the air, and the pilot heads the jet in the direction \(\mathrm{N} 45^{\circ} \mathrm{E}\) (a) Express the velocity of the wind as a vector in component form. (b) Express the velocity of the jet relative to the air as a vector in component form. (c) Find the true velocity of the jet as a vector. (d) Find the true speed and direction of the jet.

Find \(2 \mathbf{u},-3 \mathbf{v}, \mathbf{u}+\mathbf{v},\) and \(3 \mathbf{u}-4 \mathbf{v}\) for the given vectors \(\mathbf{u}\) and \(\mathbf{v} .\) $$ \mathbf{u}=\langle 0,-1\rangle, \quad \mathbf{v}=\langle- 2,0\rangle $$

Convert the polar equation to rectangular coordinates. $$ r=\frac{1}{1+\sin \theta} $$

Work A lawn mower is pushed a distance of 200 \(\mathrm{ft}\) along a horizontal path by a constant force of 50 \(\mathrm{lb}\) . The handle of the lawn mower is held at an angle of \(30^{\circ}\) from the horizontal (see the figure). Find the work done.
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