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Chapter 6: Problem 17

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

Short Answer

Expert verified
Plot the complex number \(2 \sqrt{3} - 2i\) on the complex plane at coordinates \((2 \sqrt{3}, -2)\). After simplifying the magnitude and argument computations, write the complex number in polar form using the calculated values of \(r\) and \(胃\).

Step by step solution

01

Plotting the complex number

To plot the complex number, treat the real part, \(2 \sqrt{3}\), as an x-coordinate and the imaginary part, \(-2\)(ignoring \(i\)), as the y-coordinate in a coordinate plane.
02

Computing the magnitude (r)

The magnitude of a complex number (often denoted as \(r\)) in Cartesian form (a+bi) is given by: \(r = \sqrt{a^2 + b^2}\). For our complex number, calculate \(r\) as: \(r = \sqrt{(2 \sqrt{3})^2 + (-2)^2}\). Simplify the expression to find the magnitude of the complex number.
03

Computing the argument (胃)

We compute the argument (or angle), often denoted as \(胃\), by using the formula \(胃 = \arctan \left(\frac{b}{a}\right)\) for our complex number 'a + bi'. Calculate \(胃\) using the formula as: \(胃 = \arctan \left(\frac{-2}{2 \sqrt{3}}\right)\). Simplify the expresssion to find the argument of the complex number.
04

Writing the complex number in polar form

In general, a complex number \(a+bi\)can be written in polar form as: \(r(\cos 胃 + i \sin 胃)\). Now using values of \(r\) and \(胃\), write the polar form of the complex number.

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