Equation of Straight Line in Plane/Normal Form

Theorem
Let $\mathcal L$ be a straight line such that:
 * the perpendicular distance from $\mathcal L$ to the origin is $p$
 * the angle made between that perpendicular and the $x$-axis is $\alpha$.

Then $\mathcal L$ can be defined by the equation:
 * $x \cos \alpha + y \sin \alpha = p$

Proof

 * Straight-line-normal-form.png

Let $A$ be the $x$-intercept of $\mathcal L$.

Let $B$ be the $y$-intercept of $\mathcal L$.

Let $A = \tuple {a, 0}$ and $B = \tuple {0, b}$.

From the Equation of Straight Line in Plane: Two-Intercept Form, $\mathcal L$ can be expressed in the form:
 * $(1): \quad \dfrac x a + \dfrac y a = 1$

Then:

Substituting for $a$ and $b$ in $(1)$: