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.


 * Graph-of-vertical-line.png

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:

Hence the result.