= Solution
If $b,c$ are <linearly independent>, \b[$S$ is the plane through $a$ with normal $b\times c$]. Its equation is
$$
(x-a)\cdot(b\times c)=0,
$$
so it contains the origin exactly when $a\cdot(b\times c)=0$. If $b,c$ are proportional, \b[$S$ 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.
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