 Solution

If $b,c$ are linearly independent, $S$ is the plane through $a$ with normal $b\times c$. Its equation isso it contains the origin exactly when $a\cdot (b\times c)=0$. If $b,c$ are proportional, $S$ is the line through $a$ in direction $b$, containing the origin exactly when $a$ is proportional to $b$. The linear span of two nonzero vectors has dimension one or two, so neither a single point nor all of ${\mathbb{R}}^3$ occurs.