I just wanted to check whether the projection was done against a plane or against another vector.
You have suggested its a plane which means the distance of the projection is exactly the same along some standard vector.t
It means that p1 - n*<n,p1> = p2 - n*<n,p2> = p3 where n is the normal vector to the plane.
So you have two expressions for n and since ||n|| = 1 if you get 2 of the three you get the third component.
Note that p3 is the final projected point.