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.