Finding orthogonal vector to orthonormal basis vectors

Question

If you are given an orthonormal basis vector set in a defined space are you still able to find an additional vector that is orthogonal to all the vectors included in the basis? (Assuming a non-zero vector)

Answer 1

No, by definition of a basis $\{v_i\}$, every vector $v$ can be written as $$ v = \sum_i \alpha_i v_i$$ for unique $\{\alpha_i\}$ in the underlying field. Hence $\langle v , v_i \rangle = \alpha_i$, and if $v \neq 0$ there must be at least an $i$ such that $\alpha_i \neq 0$.

(I'm assuming you want a non-zero vector. But of course the naive answer to your question, as it is phrased, would be "yes, $0$ is orthogonal to everything")