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

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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 $L$ if and only if $L'$ 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 $k$.

$\blacksquare$


Sources