Complex Modulus of Difference of Complex Numbers
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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:
\(\ds \cmod {z_1 - z_2}^2\) | \(=\) | \(\ds \cmod {z_1}^2 + \cmod {-z_2}^2 + 2 \cmod {z_1} \cmod {-z_2} \map \cos {\theta_1 - \theta_2 - \pi}\) | Complex Modulus of Sum of Complex Numbers | |||||||||||
\(\ds \) | \(=\) | \(\ds \cmod {z_1}^2 + \cmod {z_2}^2 - 2 \cmod {z_1} \cmod {z_2} \map \cos {\theta_1 - \theta_2}\) | Complex Modulus of Additive Inverse |
$\blacksquare$