Why is a vector product perpendicular?

That's because when you flip the plane the cross product is completely reversed, which means it's perpendicular to the plane. See what happens when you try to take (a×b)⋅a or (a×b)⋅b (you should get 0). If a vector is perpendicular to a basis of a plane, then it is perpendicular to that entire plane.

Try tylenol or see a professional