And the problem is that there are an infinite number of vectors perpedicular to that normal vector! However, it is true that the normal vector can be taken be lying in a plane perpedicular to the give plane.

The vector from (1,2,3) to (2,3,-1), (2-1,3-2,-1-3)= (1,1,-4), also lies in the plane. Find a vector perpendicular to both by taking the cross product. THAT will be normal to the plane you want.