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

Theorem

Let $L_1$ and $L_2$ be straight lines embedded in a cartesian plane, given in general form:

\(\ds L_1: \, \)

\(\ds l_1 x + m_1 y + n_1\)

\(=\)

\(\ds 0\)

\(\ds L_2: \, \)

\(\ds l_2 x + m_2 y + n_2\)

\(=\)

\(\ds 0\)

Then $L_1$ is perpendicular to $L_2$ if and only if:

$l_1 l_2 + m_1 m_2 = 0$

Corollary

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:

\(\ds L_1: \, \)

\(\ds y\)

\(=\)

\(\ds -\dfrac {l_1} {m_1} x + \dfrac {n_1} {m_1}\)

\(\ds L_2: \, \)

\(\ds y\)

\(=\)

\(\ds -\dfrac {l_2} {m_2} x + \dfrac {n_2} {m_2}\)

Hence the slope of $L_1$ and $L_2$ are $-\dfrac {l_1} {m_1}$ and $-\dfrac {l_2} {m_2}$ respectively.

From Condition for Straight Lines in Plane to be Perpendicular: Slope Form we have:

$-\dfrac {l_1} {m_1} = \dfrac {m_2} {l_2}$

from which the result follows.

$\blacksquare$

Sources

• 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $5$