Equation for Perpendicular Bisector of Two Points in Complex Plane/Examples/2+i, 3-2i/Standard Form

Example of Use of Equation for Perpendicular Bisector of Two Points in Complex Plane
Let $L$ be the perpendicular bisector of the straight line through $2 + i$ and $3 - 2 i$ in the complex plane.

Then $L$ can be expressed by the equation:
 * $x - 3 y = 4$

Proof
Let $z_1 = 2 + i$ and $z_2 = 3 - 2 i$.

From Equation for Perpendicular Bisector of Two Points in Complex Plane/Standard Form, $L$ can be expressed by the equation:
 * $\map \Re {z_2 - z_1} x + \map \Im {z_2 - z_1} y = \dfrac {\cmod {z_2}^2 - \cmod {z_1}^2} 2$

We have:
 * $z_2 - z_1 = 1 - 3 i$
 * $\cmod z_1^2 = 2^2 + 1^2 = 5$
 * $\cmod z_2^2 = 3^2 + 2^2 = 13$

and hence we have the equation:
 * $x - 3 y = \dfrac {13 - 5} 2 = 4$