Given the equation of a plane: ax+by+cz=D, or in vector notation , how do we find the (shortest) distance of a plane to the origin?
(When a question asks for the distance of a plane to the origin, by definition it means the shortest distance.)
One way to derive the formula is this:
Let X be the point on the plane nearest to the origin.
must be perpendicular to the plane, i.e. parallel to the normal vector .
Furthermore, X lies on the plane, hence we have
Using the formula for dot product, we can get
Since is parallel to , is either 0 or 180 degrees, hence is either 1 or -1.
Thus, we have .
The shortest distance from the point X to the origin is then
Ans: Shortest distance from point to plane is