The question states:
Let $q_1,q_2,\dots,q_n$ be an orthogonal basis of $\Bbb R^n$ and let $S = \operatorname{span}\{q_1,q_2,\dots,q_k\}$, where $1 \le k \le n-1$. Show that $S^\perp = \operatorname{span}\{q_{k+1},\dots,q_n\}$.
I am not sure what the answer should really be. I know that if $S=\operatorname{span}\{q_1,\dots,q_k\}$ then $S^\perp$ must be the span of vectors that are orthogonal to each vector in $S$. Other than using this logic I am not sure how to prove $S^\perp = \operatorname{span}\{q_{k+1},\dots,q_n\}$. Should I show that if $S$ is a subset of $\Bbb R^n$ then the union of $S$ and $S^\perp$ creates the basis in $\Bbb R^n$?