Unless I misunderstand the question, it turns out to be false unless u and v are perpendicular. Consider the following counter example:

Let the plane P be the XY plane (just to keep things simple) and let u=(1,2) and v=(1,0). These vectors span P (since they're linearly independent). Let's take x=(1,1).

Then

And

So,

So unless I'm misunderstanding or there's something missing, it's not true.