Question

What is the geometric interpretation of multiplying two complex numbers?

Answer 1

Let `z_1` and `z_2` be two complex numbers.

By rewriting in exponential form,

`{(z_1=r_1e^{i theta_1}),(z_2=r_2 e^{i theta_2}):}`

So,

#z_1 cdot z_2 =r_1e^{i theta_1}cdot r_2 e^{i theta_2} =(r_1 cdot r_2)e^{i(theta_1+theta_2)}#

Hence, the product of two complex numbers can be geometrically interpreted as the combination of the product of their absolute values (`r_1 cdot r_2`) and the sum of their angles (`theta_1+theta_2`) as shown below.

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I hope that this was clear.