Complex Modulus of Difference of Complex Numbers

Theorem

Let $z_1, z_2 \in \C$ be complex numbers.

Let $\theta_1$ and $\theta_2$ be arguments of $z_1$ and $z_2$, respectively.

Then:

$\cmod {z_1 - z_2}^2 = \cmod {z_1}^2 + \cmod {z_2}^2 - 2 \cmod {z_1} \cmod {z_2} \, \map \cos {\theta_1 - \theta_2}$

Proof

By Complex Argument of Additive Inverse, $\theta_2 + \pi$ is an argument of $-z_2$.

We have:

 $\displaystyle \cmod {z_1 - z_2}^2$ $=$ $\displaystyle \cmod {z_1}^2 + \cmod {-z_2}^2 + 2 \cmod {z_1} \cmod {-z_2} \, \map \cos {\theta_1 - \theta_2 - \pi}$ Complex Modulus of Sum of Complex Numbers $\displaystyle$ $=$ $\displaystyle \cmod {z_1}^2 + \cmod {z_2}^2 - 2 \cmod {z_1} \cmod {z_2} \, \map \cos {\theta_1 - \theta_2}$ Complex Modulus of Additive Inverse

$\blacksquare$