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Chapter 8: Problem 342

Find \(\frac{z_{1}}{z_{2}}\) in polar form. $$z_{1}=15 \operatorname{cis}\left(120^{\circ}\right) ; z_{2}=3 \operatorname{cis}\left(40^{\circ}\right)$$

Short Answer

Expert verified
\(\frac{z_{1}}{z_{2}} = 5 \operatorname{cis}(80^{\circ})\).

Step by step solution

01

Understand Polar Form Division

When dividing two complex numbers in polar form, \(\frac{z_1}{z_2} = \frac{r_1}{r_2} \operatorname{cis}(\theta_1 - \theta_2)\). In this form, \(r_1\) and \(r_2\) are the magnitudes, while \(\theta_1\) and \(\theta_2\) are the angles.
02

Calculate the Magnitude

Identify the magnitudes of the complex numbers. \(r_1 = 15\) and \(r_2 = 3\). Compute \(\frac{r_1}{r_2} = \frac{15}{3} = 5\).
03

Subtract the Angles

Identify the angles: \(\theta_1 = 120^{\circ}\) and \(\theta_2 = 40^{\circ}\). Compute \( \theta_1 - \theta_2 = 120^{\circ} - 40^{\circ} = 80^{\circ}\).
04

Write the Result in Polar Form

Combine the results from Step 2 and Step 3. The polar form for \(\frac{z_{1}}{z_{2}}\) is \(5\operatorname{cis}(80^{\circ})\).

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Key Concepts

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

Polar Form
Polar form is a way to express complex numbers. Instead of writing them as a sum of a real and an imaginary part, complex numbers in polar form use a magnitude and a direction (angle). This is particularly useful when multiplying or dividing complex numbers.

Polar form looks like this: \(r \operatorname{cis}(\theta)\), where:
  • \(r\) is the magnitude or the distance from the origin to the point on a complex plane.
  • \(\theta\) is the angle formed with the positive x-axis.
  • \(\operatorname{cis}(\theta)\) represents \(\cos(\theta) + i \sin(\theta)\).
The polar form provides a compact way to handle complex numbers, making mathematical operations straightforward, especially when dealing with powers and roots.
Magnitude
Magnitude refers to the absolute value or the distance of a complex number from the origin on the complex plane. It is like the length of a vector. When in polar form, a complex number's magnitude is denoted as \(r\).

For a complex number in the form \(a + bi\), the magnitude \(r\) can be calculated using the formula: \[r = \sqrt{a^2 + b^2}\]In the given exercise, we had two magnitudes: \(r_1 = 15\) and \(r_2 = 3\). To compute the magnitude of the division \(\frac{z_1}{z_2}\), we simply divide the magnitudes: \[\frac{r_1}{r_2} = \frac{15}{3} = 5\]Understanding magnitude helps in visualizing the size or distance a complex number represents on the complex plane.
Angle Subtraction
Angle subtraction is a crucial step in dividing complex numbers in polar form. When dividing two complex numbers, the operation involves subtracting their respective angles, not dividing them.

In the format \(\frac{z_1}{z_2} = \frac{r_1}{r_2} \operatorname{cis}(\theta_1 - \theta_2)\), the process involves:
  • Identifying the angles: \(\theta_1\) and \(\theta_2\).
  • Subtracting these angles: \(\theta_1 - \theta_2\).
In our exercise, \(\theta_1 = 120^{\circ}\) and \(\theta_2 = 40^{\circ}\). Subtracting these gives us:\[120^{\circ} - 40^{\circ} = 80^{\circ}\]The angle subtraction is simple: just subtract the second angle from the first to adjust the directionality of the resulting complex number.
Division of Complex Numbers
Division of complex numbers can be simplified by using their polar form. Unlike division in general arithmetic, it combines two actions: dividing magnitudes and subtracting angles.

The formula to divide two complex numbers in polar form is:\[\frac{z_1}{z_2} = \frac{r_1}{r_2} \operatorname{cis}(\theta_1 - \theta_2)\]Here's how it works:
  • Calculate the magnitude of the division: \(\frac{r_1}{r_2}\).
  • Subtract the angle of the second number from the first: \(\theta_1 - \theta_2\).
In our example, we ended with \(5 \operatorname{cis}(80^{\circ})\) because we divided magnitude \(15/3\) and subtracted angles \(120^{\circ} - 40^{\circ}\).

By handling the polar form, division becomes a much more intuitive and streamlined process, avoiding the more complex arithmetic of components.

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Most popular questions from this chapter

Plot the point with polar coordinates \(\left(5,-\frac{2 \pi}{3}\right)\)

A man starts walking from home and walks 3 miles at \(20^{\circ}\) north of west, then 5 miles at \(10^{\circ}\) west of south, then 4 miles at \(15^{\circ}\) north of east. If he walked straight home, how far would he have to the walk, and in what direction?

Convert \(\left(2, \frac{\pi}{3}\right)\) to rectangular coordinates.

For the following exercises, find a unit vector in the same direction as the given vector. $$a=8 i-6 j$$

Find the absolute value of the complex number \(5-9 i\).
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