Is it always possible to find a vector perpendicular to two given vectors in a general inner product space?

Question

In short, given an inner product space $X$ , for any $x,y \in X$ does there always exist a nontrivial $z \in X$ so that $<x,z> = 0$ and $<y,z> = 0$ ?

Answer 1

Not necessarily.

If you take the space $\mathbb{R}^2$ with the dot product as the inner product and $x=(0,1)$ and $y=(1,0)$, then $\nexists$ any such non-trivial $z$ which will satisfy the condition.