Question

If the vectors b, c, d are non-coplanar, then prove that the vector (a x b) x (c x d) + (a x c) x (d x b) + (a x d) x (b x c) is a vector parallel to a.

Answer 1

Vector b, c, d are given to be non-coplanar and hence [b, c, d] ≠ 0 Any other vector a can be expressed as a linear combination of b, c, and d so that we have

Hence, by adding the three results and using above information, we get

This is certainly in the direction of vector a.