Condition for Straight Lines in Plane to be Perpendicular/Slope Form/Proof 3

Theorem

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:

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

Proof

Let $\psi$ be the angle between $L_1$ and $L_2$

From Angle between Straight Lines in Plane:

$\psi = \arctan \dfrac {m_1 - m_2} {1 + m_1 m_2}$

When $L_1$ and $L_2$ are perpendicular:

$\psi = \dfrac \pi 2$

by definition.

From Tangent of Right Angle $\tan \dfrac \pi 2$ is undefined.

This happens if and only if $1 + m_1 m_2 = 0$.

The result follows.

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {III}$. Analytical Geometry: The Straight Line: The angle between two lines