Question

plot the product of the complex numbers on the plane below.

$(2e^{315^{circ }i})(2e^{210^{circ }i})$

answer

points can be moved by dragging or using the arrow keys. if using the arrow keys, note that the right/left arrow will increase/decrease the radius and the up/down arrow will increase/decrease the angle.

Explanation:

Step1: Multiply the magnitudes

When multiplying complex numbers in polar form \(z_1 = r_1e^{i\theta_1}\) and \(z_2=r_2e^{i\theta_2}\), the magnitude of the product \(z = z_1z_2\) is \(r=r_1r_2\). Here, \(r_1 = 2\) and \(r_2=2\), so \(r=2\times2 = 4\).

Step2: Add the angles

The angle of the product \(z\) is \(\theta=\theta_1+\theta_2\). Given \(\theta_1 = 315^{\circ}\) and \(\theta_2=210^{\circ}\), then \(\theta=315^{\circ}+210^{\circ}=525^{\circ}\). Since angles in the complex - plane are periodic with period \(360^{\circ}\), we find the equivalent angle \(\theta = 525^{\circ}-360^{\circ}=165^{\circ}\)

Answer:

The complex number \((2e^{315^{\circ}i})(2e^{210^{\circ}i})\) has a magnitude of \(4\) and an angle of \(165^{\circ}\). So, on the polar - plane, you should move the point to a circle corresponding to radius \(4\) (using the right/left arrow keys to adjust the radius) and then use the up/down arrow keys to set the angle to \(165^{\circ}\)