Fun with Math!: Distance from a Point to a Line

If we have a point S and a line l that passes through point P and is parallel to vector v, then the shortest distance from point S to the line l is $d=\frac{|\overrightarrow{PS}\times\textbf{v}|}{|\textbf{v}|}$ .

The Forming of the Formula

Let's take a look on the process to get the formula. Let say point S₁ is the point on the line l which gives the shortest distance from point S to the line. As such, a right triangle PS₁S is formed. The distance of SS₁ (the shortest distance) can be found using the Pythagoras theorem:

$|\overrightarrow{PS}|\sin\theta\\ =|\overrightarrow{PS}|\sin\theta\;\frac{|\textbf{v}|}{|\textbf{v}|}\\ =\frac{|\overrightarrow{PS}||\textbf{v}|\sin\theta}{|\textbf{v}|}\\ =\frac{|\overrightarrow{PS}||\textbf{v}||\sin\theta|}{|\textbf{v}|}\;\textrm{ since } \sin\theta\geqslant 0\textrm{ for }0\leqslant \theta\leqslant\frac{\pi}{2}\\ =\frac{|\overrightarrow{PS}\times\textbf{v}|}{|\textbf{v}|}\;\textrm{ according to the definition of the cross product}$ .
The graph below is meant to show the distance from a point to a line. Move point P or T to move the line; move point S to see the changes of the right triangle. Either point will give changes to the distance. The coordinates of points P, S, initial point of v and terminal point of v are given. You may check if the vectors in the calculation are correct (sorry for the minor rounding errors inside).

Betty, Created with GeoGebra and Thomas Calculus, Pearson.

Fun with Math!

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Saturday, May 9, 2020

Distance from a Point to a Line

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