Existence of a Vector in an Inner Product Space 
Let V be an inner product space and let $v \neq v'$ be vectors in $V$ . Show that there exists a vector $w \in V$ satisfying $\langle v,w\rangle\:\neq \:\langle v',w\rangle$.

As $v-v'\neq 0$, we get 
$$\langle v-v',v-v'\rangle >0$$
which implies 
$$\langle v,v-v'\rangle -\langle v',v-v'\rangle >0$$
$$\langle v,v-v'\rangle >\langle v',v-v'\rangle $$
Let $w=v-v'$. I am not sure if my argument is correct. I appreciate any help.

$v-v'\ne 0\,$ is assumed. And $w$ is wanted with $\langle v-v',w\rangle\neq 0\,$. 
Then $w=v-v'$ does the job because an inner product is positive definite.
This is merely a short reformulation of the correct argument given in the OP.
