Equation of Straight Line in Plane/Normal Form/Proof 2
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Theorem
Let $\LL$ be a straight line such that:
- the perpendicular distance from $\LL$ to the origin is $p$
- the angle made between that perpendicular and the $x$-axis is $\alpha$.
Then $\LL$ can be defined by the equation:
- $x \cos \alpha + y \sin \alpha = p$
Proof
Let $P = \tuple {x, y}$ be an arbitrary point on $\LL$.
Let $O$ be the origin of the Cartesian plane in which $\LL$ is embedded.
Let $PQ$ be the perpendicular dropped from $P$ to the $x$-axis.
Let $QS$ be the perpendicular dropped from $Q$ to the line $ON$.
Let $PR$ be the perpendicular dropped from $P$ to the line $QS$.
By definition of cosine:
- $OS = OQ \cos \alpha$
By definition of sine:
- $PR = PQ \sin \alpha$
Then:
\(\ds p\) | \(=\) | \(\ds ON\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds OS + SN\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds OS + PR\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds OQ \cos \alpha + PQ \sin \alpha\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x \cos \alpha + y \sin \alpha\) |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {III}$. Analytical Geometry: The Straight Line: Equation of a Straight Line: $\tuple {p, \alpha}$ form