# Equation of Vertical Line

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## 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.

$\blacksquare$