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:
 * $\left\vert{z_1-z_2}\right\vert^2 = \left\vert{z_1}\right\vert^2 + \left\vert{z_2}\right\vert^2 - 2 \left\vert{z_1}\right\vert\left\vert{z_2}\right\vert \cos(\theta_1-\theta_2)$

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