# Equation of Vertical Line

## Theorem

Let $\mathcal L$ be a vertical line embedded in the Cartesian plane $\mathcal C$.

Then the equation of $\mathcal L$ can be given by:

$x = a$

where $\tuple {a, 0}$ is the point at which $\mathcal L$ intersects the $x$-axis.

## Proof

From the Normal Form of Equation of Straight Line in Plane, a general straight line can be expressed in the form:

$x \cos \alpha + y \sin \alpha = p$

where:

$p$ is the length of a perpendicular $\mathcal P$ from $\mathcal L$ to the origin.
$\alpha$ is the angle made between $\mathcal P$ and the $x$-axis.

As $\mathcal L$ is vertical, then by definition $\mathcal P$ is horizontal.

By definition, the horizontal line through the origin is the $x$-axis itself.

Thus $\alpha = 0$ and $p = a$

Hence the equation of $\mathcal L$ becomes:

 $\displaystyle x \cos 0 + y \sin 0$ $=$ $\displaystyle a$ $\displaystyle \leadsto \ \$ $\displaystyle x$ $=$ $\displaystyle a$ Sine of Zero is Zero, Cosine of Zero is One

Hence the result.

$\blacksquare$