Q: Prove that if and only if and have the same direction.
A: Well, if the same direction of , then must be a multiple of . Thus, where (I am not sure if I lost any generality by choosing ). From here I expanded and ended up with some messy algebra and not the result I was looking for.
In the other direction I started with and worked my way down to and figured I have to introduce the Cauchy-Schawrz Ineguality here, but I am not sure how.
I am having a hard time formalizing this proof. All I have is a bunch of dead und scratch work. It seems that this proof should flow with ease, because, geometrically, it is pretty obviouse.
I would appreciate some guidence.