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.

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.

$\blacksquare$

Also see

• Equation of Horizontal Line