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Chapter 6: Problem 22

Plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. $$3 \sqrt{2}-3 i \sqrt{2}$$

Short Answer

Expert verified
The plotted point of the complex number on the complex plane is \( (3 \sqrt{2}, -3 \sqrt{2}) \). The polar form of the complex number has an absolute value of 6 and an argument of -45掳 or -\(\frac{\pi}{4}\) radian and is written as \(6[cos(-45掳) + i sin(-45掳)]\) or \(6[cos(-\frac{\pi}{4}) + i sin(-\frac{\pi}{4})]\).

Step by step solution

01

Plot the Complex Number

Plot the point \((3 \sqrt{2}, -3 \sqrt{2})\) on the complex plane. The real part \(3 \sqrt{2}\) will be on the X-axis, while the imaginary part \(-3 \sqrt{2}\) will be on the Y-axis.
02

Find the Absolute Value (r)

Find the absolute value of the complex number by using the formula \(r = \sqrt{(a)^2 + (b)^2}\). Substituting the given \(a = 3 \sqrt{2}\) and \(b = -3 \sqrt{2}\), the absolute value \(r\) is 6.
03

Calculate the Argument (\(\Theta\))

The argument of the complex number, often denoted \(\Theta\), is calculated using the formula \(\Theta = atan2(b, a)\). For the given complex number, \(\Theta = atan2(-3 \sqrt{2}, 3 \sqrt{2})\), which simplifies to \(-45掳\) or \(-\frac{\pi}{4}\) radian.
04

Write the Polar Form

Substitute the absolute value and the argument into the polar form equation \(r(cos(\Theta) + i sin(\Theta))\). In this case, the polar form will be \(6[cos(-45掳) + i sin(-45掳)]\) or \(6[cos(-\frac{\pi}{4}) + i sin(-\frac{\pi}{4})]\).

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