You can put this solution on YOUR website! You can prove the converse, that the vector normal (you use perpendicular, not exactly the same) to a plane is unique in direction. Then any line parallel to the vector thru a chosen point is normal to the plane, and not normal to any plane not parallel to the given plane.
As an alternative, choose a point on a line in 3-space. Find 2 lines not 180º apart perpendicular to the 1st line. Those 2 lines define a plane.