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:

Also see

 * Complex Modulus of Sum of Complex Numbers