Plane Intersections

Given a point $p_0$ on a plane as well as a unit vector normal to the place $n$.
Any point $x$ on the plane satisfies $n\cdot (x-p_0)=0$ meaning that the vector from x to $p_0$ where we know $p_0$ is on the plane must be tangent to the plane. Therefore, the dot product must be 0 as there is an angle of 90° between the two vectors.

The plane could also be given as $ax+by+cz+d=0$ where $n=(a,b,c)$ and $d=n\cdot p_0$ which is how far the plane is from the origin along its normal.