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.
$\blacksquare$