To prove coplanarity of three vectors $a$, $b$, and $c$ we use the scalar triple product $a\cdot (b\times c)=0$. This is because $b\times c$ is a vector perpendicular to plane containing $b$ and $c$and also perpendicular to vector $a$. And hence $a$, $b$, and $c$ are coplanar.
My question is this needn't necessary be true, right? What if $a$ is a vector on a plane parallel to plane containing $b$ and $c$? Then $b\times c$ will be perpendicular to $a$ and the triple product zero.