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 $\mu_1$

$\blacksquare$

Sources

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