Can $v$ be a linear combination of nonzero independent vectors it is orthogonal to?

Question

Let $u,v,w$ be nonzero vectors in $\mathbb{C}^{n} $ such that $ \langle w,u \rangle = \langle w,v \rangle = 0$. Suppose that $u,v$ are linearly independent.
Is it possible that $w \in span(u,v)$ ? Proof?

Answer 1

Suppose $w \in \operatorname{span}{\{u,v\}}$. Then $w=au+bv$ for some scalars $a,b$. Then $$ \langle w,w \rangle = \langle w, au+bv \rangle = a\langle w,u \rangle + b \langle w,v \rangle = 0, $$ so if $\langle \cdot,\cdot \rangle$ is a true inner product (i.e. nondegenerate), $w=0$.