How to check if a point is in the direction of the normal of a plane?

Question

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?

Answer 1

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 $n$.

Answer 2

You can check if the point is on the plane by

$$ d = \vec{n} \cdot (\vec{a}-\vec{p}) = 0 $$

where $\cdot$ is the vector dot product, and $d$ is the distance of the point to the plane in the direction of the normal.