Finding if a point is in front or behind a plane

The question is how to find, if specified point is behind a plane. And by plane I rather mean a point with a given direction.

Let's say the plane is a point A with direction dirA, and the point we want to check is B

My thoughts were to check if dot product of dirA and a vector, that represents the length from the "plane" to the point B. But the thing is that vector would need to be perpendicular to the plane, and I got no idea how to get that vector.

Is my concept right? If it is, how do I get the vector I need?