Lecture 6: Distance between Point and Plane

In Euclidean space, the point on a plane ax+by+cz=d that is closest to the origin has the Cartesian coordinates (x,y,z), where

x=ad/a^{2}+b^{2}+c^{2}, y=bd/a^{2}+b^{2}+c^{2}, z=cd/a^{2}+b^{2}+c^{2}.

The distance between the origin and point (x,y,z) is sqrt (x^{2}+y^{2}+z^{2}).

If what is desired is the distance from a point not at the origin to the nearest point on a plane, this can be found by a change of variables that moves the origin to coincide with the given point.

Suppose we wish to find the nearest point on a plane to the point ( X0,Y0,Z0) , where the plane is given by aX+bY+cZ=D. We define x=X-X0, y=Y-Y0, z=Z-Z0 , and d=D-aX0-bY0-cZ0, to obtain ax+by+cz=d ax+by+cz=d as the plane expressed in terms of the transformed variables. Now the problem has become one of finding the nearest point on this plane to the origin, and its distance from the origin. The point on the plane in terms of the original coordinates can be found from this point using the above relationships between x and X, between y and Y, and between z and Z; the distance in terms of the original coordinates is the same as the distance in terms of the revised coordinates.