I am stuck with a proof of which I think that I have an idea of how to solve it.
I have given two vectors with . I know that and .
Now I am supposed to show a couple of things and what I am left to do is to prove that there is only one s.t. and .
OK, now here is what I have done. I know from the Schwarz inequality that is possible in general if I chose the right z. I get this same equality if I set the given definition of d into my other constraint. Also should I be able to solve for x since I have one unknown z and three requirements for z. However, whenever I solve for z I get some nonsence. I am not exactly sure about what I am doing wrong or how else I can show that there is only one z for that all these requirements hold. Can one of you help me. Even a hint would be more than enough. Thanks a ton! Cheers.
I am not sure if I got your hint. However, here is what I have done.
First, for convenience, I defined: and what should, in my opinion, only "shift" the problem but not change it.
I know that: and therefore . Since , I know that and can therefore not be any other value.
Q.E.D.? I'm still confused.
You have to use the Cauchy-Schwarz inequality here to conclude that x' and z must be collinear:
C-S inequality says that with equality iff z and x' are collinear. In this case we have equality . So there is a , such that .
From this it follows that . So is the only possible value for z.