I have a plane whose normal $n = (a,b,c)$ is known, also the point $x_0,y_0,z_0$ that the plane passes through is given. Now I am given a 2D vector on this plane $u = (u_1,u_2)$ which is pointing from $x_0,y_0,z_0$ to a new point $X$. $u$ is defined according to the coordinate system which is captured by $v,w$ coordinates, that are projections of real world unit vectors in 3D. How do I find the 3D coordinate of the new point $X$?
