Let $Π_1$ and $Π_2$ be two planes determined by the pairs of vectors $\vec a,\vec b$ and $\vec c,\vec d$ respectively. If the planes $Π_1$ and $Π_2$ are parallel, then

$(\vec a×\vec b)×(\vec c×\vec d) =\vec 0$

Two planes are parallel, if their normals are parallel.

$∴Π_1$ and $Π_2$ are parallel

$⇒\vec a×\vec b$ and $\vec c×\vec d$ are parallel

$⇒(\vec a×\vec b)×(\vec c×\vec d) =\vec 0$