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

If we want to find the shortest distance from a point S to the plane containing point P and perpendicualr to the vector n, we just need to use the formula
$d=\frac{|\overrightarrow{PS}\cdot\textbf{n}|}{|\textbf{n}|}$ .

The Forming of the Formula

How do we get the formula? First, let's say point S' on tha plane has the shortest distance to the point S. Then, the line segment connecting S and S' must be perpendicular to the plane. Then, we form a right triangle PS'S. If the angle PSS' is A, then the length SS' (distance from S to S') is

$|\overrightarrow{PS}|\cos A\\ =|\overrightarrow{PS}|\cos A\frac{|\textbf{n}|}{|\textbf{n}|}\\ =\frac{|\overrightarrow{PS}|\cos A|\textbf{n}|}{|\textbf{n}|}\\ =\frac{|\overrightarrow{PS}||\textbf{n}|\cos A}{|\textbf{n}|}\\ =\frac{\overrightarrow{PS}\cdot\textbf{n}}{|\textbf{n}|}\\ =\frac{|\overrightarrow{PS}\cdot\textbf{n}|}{|\textbf{n}|} \textrm{ because }\cos A\geqslant 0 \textrm{ for } 0\leqslant A\leqslant\frac{\pi}{2}$ .

A further explanation on the equation above. Since n could be facing the opposite direction that cause the dot product to be negative, we put in the absolute sign to ensure we get non-negative number for distance.

You may try to move point P or D to move the plane, P or S to observe the changes of the distance.

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 Plane

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