# Product of Complex Number with Conjugate in Exponential Form

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## Theorem

Let $z_1$ and $z_2$ be complex numbers.

Then:

$\overline {z_1} z_2 = \cmod {z_1} \, \cmod {z_2} e^{i \theta}$

where:

$\overline {z_1}$ denotes the complex conjugate of $z_1$
$\cmod {z_1}$ denotes the complex modulus of $z_1$
$\theta$ denotes the angle from $z_1$ to $z_2$, measured in the positive direction.

## Proof

 $\ds \overline {z_1} z_2$ $=$ $\ds \paren {z_1 \circ z_2} + i \paren {z_1 \times z_2}$ Product of Complex Number with Conjugate by Dot and Cross Product $\ds$ $=$ $\ds \cmod {z_1} \, \cmod {z_2} \cos \theta + i \paren {z_1 \times z_2}$ Definition 2 of Dot Product $\ds$ $=$ $\ds \cmod {z_1} \, \cmod {z_2} \cos \theta + i \cmod {z_1} \, \cmod {z_2} \sin \theta$ Definition 2 of Vector Cross Product $\ds$ $=$ $\ds \cmod {z_1} \, \cmod {z_2} \paren {\cos \theta + i \sin \theta}$ $\ds$ $=$ $\ds \cmod {z_1} \, \cmod {z_2} e^{i \theta}$ Euler's Formula

$\blacksquare$