Projection of a point on a plane

Let consider the plane (p) of equation


and a point M(u,v,w)
We look for the point $ N =(x_0,y_0,z_0) $, 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

$$\left<br />
{\begin{array}{c}<br />
x=u+at\\<br />
y=v+bt\\<br />
z=w+ct<br />

Now we want the value of $ t_0 $ for with a point of this line is on the plane.


so we have

$$\left<br />
{\begin{array}{c}<br />
x_0=u+at_0\\<br />
y_0=v+bt_0\\<br />
z_0=w+ct_0<br />

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

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