In an elementary linear algebra (or calculus) course, we would solve this problem as follows. First, we would need two vectors parallel to the plane. If lies in the plane, then so and
so and are parallel to the plane. We then compute the normal vector
and compute the projection of the position vector for the point onto This gives the vector
Now, this vector is parallel to so it’s perpendicular to the plane. Subtracting it from gives a vector parallel to the plane, and this is the position vector for the point we seek.
so the closest point is We weren’t asked for it, but note that if we wanted the distance from the point to the plane, this is given by