# Equation of Vertical Line

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## 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.

## 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:

\(\ds x \cos 0 + y \sin 0\) | \(=\) | \(\ds a\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds a\) | Sine of Zero is Zero, Cosine of Zero is One |

Hence the result.

$\blacksquare$