Product of Complex Number with Conjugate in Exponential Form

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.

$\ds \overline {z_1} z_2$

$=$

$\ds \paren {z_1 \circ z_2} + i \paren {z_1 \times z_2}$

$\ds $

$=$

$\ds \cmod {z_1} \, \cmod {z_2} \cos \theta + i \paren {z_1 \times z_2}$

Definition 2 of Dot Product

$\ds $

$=$

$\ds \cmod {z_1} \, \cmod {z_2} \cos \theta + i \cmod {z_1} \, \cmod {z_2} \sin \theta$

$\ds $

$=$

$\ds \cmod {z_1} \, \cmod {z_2} \paren {\cos \theta + i \sin \theta}$

$\ds $

$=$

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

$\blacksquare$