The pairs a→,b→ and c→,d→ each determine a plane. Then the planes are parallel if

a→×b→ is a vector perpendicular to the plane  a→ and b→ Similarly c→×d→ is a vector perpendicular to the plane containing c→ and d→. Thus, the two planes will be parallel if their normals, i.e., a→×b→ and c→×d→ are parallel. Thus(a→×b→)×(c→×d→))=0→