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Chapter 6: Problem 21

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

Short Answer

Expert verified
The complex number \(-3 \sqrt{2}-3 i \sqrt{3}\) corresponds to the point (-3\(\sqrt{2}\), -3\(\sqrt{3}\) on the complex plane. Its magnitude is \(3\sqrt{5}\) and its argument is \(2\pi/3\). Therefore, the polar form of the number is \(3\sqrt{5}(cos(2\pi/3) + i sin(2\pi/3))\).

Step by step solution

01

Plot the Complex Number

First, identify the real and imaginary parts of the complex number. The real part is -3\(\sqrt{2}\), and the imaginary part is -3\(\sqrt{3}\). The complex number represents a point on the complex plane, where the x-coordinate is the real part and the y-coordinate is the imaginary part. Therefore, the point on the complex plane to plot is (-3\(\sqrt{2}\), -3\(\sqrt{3}\).
02

Calculate the Magnitude

In polar form, the magnitude of a complex number, denoted as r, is calculated using the formula \(r = \sqrt{a^2 + b^2}\), where a and b are the real and imaginary parts of the number respectively. Here, \(r = \sqrt{(-3\sqrt{2})^2 + (-3\sqrt{3})^2} = \sqrt{18+27} = \sqrt{45} = 3\sqrt{5}\).
03

Calculate the Argument

The argument, denoted as \(\theta\), is the angle made with the positive x-axis (real line). It can be calculated using the formula \(\theta = arctan(b/a)\) but considering the signs of a and b, because the arctan only returns values from -\(\pi/2\) to \(\pi/2\), so it must be adjusted based on the quadrant. Here, a=-3\(\sqrt{2}\) and b=-3\(\sqrt{3}\) are negative, so the complex number is in the third quadrant, and \(\theta = arctan(-3\sqrt{3}/-3\sqrt{2})+\pi = 2\pi/3\).
04

Write the Complex Number in Polar Form

The polar form of a complex number is represented as \(r(cos\(\theta\) + isin\(\theta\))\). Substituting the values for r and \(\theta\) derived above, the polar form of the complex number is \(3\sqrt{5}(cos(2\pi/3) + i sin(2\pi/3))\).

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Most popular questions from this chapter

Write an equation in point-slope form and general form for the line passing through (-2,5) and perpendicular to the line whose equation is \(x-4 y+8=0\) (Section \(1.5,\) Example 2 )

Determine whether v and w are parallel, orthogonal, or neither. $$\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}, \quad \mathbf{w}=6 \mathbf{i}+\frac{18}{5} \mathbf{j}$$

How do you determine the work done by a force F in moving an object from \(A\) to \(B\) when the direction of the force is not along the line of motion?

Use the dot product to determine whether v and w are orthogonal. $$\mathbf{v}=5 \mathbf{i}-5 \mathbf{j}, \quad \mathbf{w}=\mathbf{i}-\mathbf{j}$$

Verify the identity: $$\sin ^{2} x \tan ^{2} x+\cos ^{2} x \tan ^{2} x=\sec ^{2} x-1$$
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