Condition for Straight Lines in Plane to be Perpendicular/General Equation/Corollary

Theorem
Let $L$ be a straight line in the Cartesian plane.

Let $L$ be described by the general equation for the straight line:
 * $l x + m y + n = 0$

Then the straight line $L'$ is perpendicular to $L$ $L'$ can be expressed in the form:


 * $m x - l y = k$

Proof
From the general equation for the straight line, $L$ can be expressed as:


 * $y = -\dfrac l m x + \dfrac n m$

Hence the slope of $L$ is $-\dfrac l m$.

Let $L'$ be perpendicular to $L$.

From Condition for Straight Lines in Plane to be Perpendicular, the slope of $L'$ is $\dfrac m l$.

Hence $L'$ has the equation:


 * $y = \dfrac m l x + r$

for some $r \in \R$.

Hence by multiplying by $l$ and rearranging:
 * $m x - l y = -l r$

The result follows by replacing the arbitrary $-l r$ with the equally arbitrary $k$.