For the sake of clear explanation, it would probably be helpful to note why it is that the choice of such a vector does not lose you any generality. Alternatively, you could just consider a vector of arbitrary length?
Also, your last part is right but it seems as though, at this point, it is merely a conjecture based on the initial geometric argument. Perhaps it would be better to use a more abstract algebraic argument based on a more sophisticated (and more abstract) understanding of vector principles?
Smaug seemed far more well-versed with vector terminology etc.. so I will leave it to him to formalize your language (seems fine to me
).