I have a plane, defined by a normal vector $n$ and a point $p$. I also have a point $a = (x, y, z)$. Based on this information, how do I know if the point $a$ exists somewhere past the plane in the direction of the normal?
Take the dot product $n \cdot (a-p)$: if the sign is positive then $a$ exists somewhere past the plane in the direction of the normal $n$. If such dot product is zero then $a$ is in the plane and in the case the dot product is negative is in the opposite side of the plane w.r.t the normal $n$.