# Condition for Straight Lines in Plane to be Perpendicular/Slope Form/Proof 1

## 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 $\mu_1 = \tan \phi$.

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

 $\ds \mu_2$ $=$ $\ds \tan {\phi + \dfrac \pi 2}$ Definition of Perpendicular $\ds$ $=$ $\ds -\cot \phi$ Tangent of Angle plus Right Angle $\ds$ $=$ $\ds -\dfrac 1 {\tan \phi}$ Definition of Cotangent of Angle $\ds$ $=$ $\ds -\dfrac 1 {\mu_1}$ Definition of $\mu_1$

$\blacksquare$