Condition for Straight Lines in Plane to be Perpendicular

Theorem

General Equation

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:

$l_1 l_2 + m_1 m_2 = 0$

Slope Form

Let $L_1$ and $L_2$ be straight lines in the Cartesian plane.

Let the slope of $L_1$ and $L_2$ be $\mu_1$ and $\mu_2$ respectively.

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

$\mu_1 = -\dfrac 1 {\mu_2}$

Examples

Arbitrary Example $1$

Let $\LL_1$ be the straight line whose equation in general form is given as:

$3 x - 4 y = 7$

Let $\LL_2$ be the straight line perpendicular to $\LL_1$ which passes through the point $\tuple {1, 2}$.

The equation for $\LL_2$ is:

$4 x + 3 y = 10$