A linear system $w = c_1 v_1 + \dots + c_k v_k$ has no solution for the scalars $c_i$. What does this algebraic failure reveal geometrically about $w$ relative to the set $\{v_1, \dots, v_k\}$?

C. The vector $w$ lies entirely outside the subspace spanned by the set

Insolubility means no weighting of the vectors can reach $w$, placing it outside their span. It need not be orthogonal or perpendicular to the subspace, merely non-coplanar with the reach of the set.