# Projection of a point on a plane

Submitted by Structure on Sat, 03/22/2008 - 20:49.

Let consider the plane (p) of equation

and a point M(u,v,w)

We look for the point , the projection of M on the plane.Normal of the plane is the vector (a,b,c) so line by M(u,v,w) of equations

is the line perpendicular on the plane.

so we have for a point on this line parametric equations

Now we want the value of for with a point of this line is on the plane.

so we have

We can find the distance from the point M(u.v.w) to the plane ax+by+cz+d=0.

The square of this distance is

so we have