Complex Numbers: Orthogonality/Right Angle Condition

How do we know if two segments, $AB$ and $CD$, are perpendicular if they are not touching?

Assign complex numbers $a=A, b=B, c=C, d=D$.

Since moving two segments around doesn't change the angle between them, we translate $AB$ so that A is at the origin and $CD$ so C is at the origin.

After this transformation, $b'=b-a, d'=d-c$.

This means that $d'*xi=b'$ for some real number x, because multiplying by i rotates a number 90 degrees! (And so does multiplying a number by 2i or any multiple of i).

So two segments $AB, CD$ are perpendicular if and only if $\frac{b-a}{d-c}=ki$ for some number k.

We can rewrite this condition using the conjugate. Notice that if z is pure imaginary, $z=-Z$ where Z is the conjugate of z.

Therefore, $\frac{b-a}{d-c}=-\frac{B-A}{D-C}$ (capital letters denote conjugates of lower-case letters).

This is an equivalent condition for segments to be perpendicular.