You need to find the vector normal to the plane going through A. It will intersect the plane at point P, and have its tip at point A. Take the negative of this vector. Put its tail at P, and then the tip will be at the reflection point.
The only difference between what I have said and what you have said is that I claim you need a normal vector. I don't think this will work if the vector you find is not normal to the plane.