For three vectors u, v and w show that:

Question

(u×v).[(v×w)×(w×u)]= [u.(v×w)]^2 .

Answer 1

You have probably already seen that

Ax(wxu) = w(A.u)-u(A.w)
Now let A = vxw
Then
(vxw)x(wxu) = w(uxvxw)
So,
(uxv).(vxw)x(wxu) = (uxv).w(u.vxw)
= (uxv.w)(u.vxw)
= (u.vxw)^2

There are other proofs online