Perpendicular Distance from Straight Line in Plane to Point
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Theorem
General Form
Let $\LL$ be a straight line embedded in a cartesian plane, given by the equation:
- $a x + b y + c = 0$
Let $P$ be a point in the cartesian plane whose coordinates are given by:
- $P = \tuple {x_0, y_0}$
Then the perpendicular distance $d$ from $P$ to $\LL$ is given by:
- $d = \dfrac {\size {a x_0 + b y_0 + c} } {\sqrt {a^2 + b^2} }$
Normal Form
Let $\LL$ be a straight line in the Cartesian plane.
Let $\LL$ be expressed in normal form:
- $x \cos \alpha + y \sin \alpha = p$
Let $P$ be a point in the cartesian plane whose coordinates are given by:
- $P = \tuple {x_0, y_0}$
Then the perpendicular distance $d$ from $P$ to $\LL$ is given by:
- $\pm d = x_0 \cos \alpha = y_0 \sin \alpha - p$
where $\pm$ depends on whether $P$ is on the same side of $\LL$ as the origin $O$.