What is the geometric interpretation of multiplying two complex numbers?

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What is the geometric interpretation of multiplying two complex numbers?

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Answer 1 · 2014-12-12T07:05:37

Let $z_{1}$ $z_1$ and $z_{2}$ $z_2$ 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.

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What is the geometric interpretation of multiplying two complex numbers?

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