Equation of Vertical Line

Theorem
Let $\LL$ be a vertical line embedded in the Cartesian plane $\CC$.

Then the equation of $\LL$ can be given by:
 * $x = a$

where $\tuple {a, 0}$ is the point at which $\LL$ 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 $\PP$ from $\LL$ to the origin.
 * $\alpha$ is the angle made between $\PP$ and the $x$-axis.

As $\LL$ is vertical, then by definition $\PP$ 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 $\LL$ becomes:

Hence the result.