Show that (a ^ b)^2 . (c ^ d) = (a . c)(b . d) - (a . d(b . c
Calculations:
(a ^ b) = Eijk*e^(i)*ai*bj, where i,j,k are subscripts, ^ = permutation symbold, and ^(i) = raise to the power of i
(c ^ d) = Epqk*e^(i)*ck*dp, where p,q,k are subscripts, ^ = permutation symbold, and ^(i) = raise to the power of i
(a ^ b) . (c ^ d) = (Eijk*e^(i)*ai*bj)(Epqk*e^(i)*ck*dp)
EijkEpqk = a three by three matrix:
dip diq dik
djp djq djk
dkp dkq dkk, where d = delta
= ddipddjq - djpdiq
(a ^ b) . (c ^ d = (dipdjq - djpdiq)*ai*bj*ck*dp
= (dipdjq*ai*bq*ck*dp) - (djpddiq*ai*bj*ck*dp)
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