Vector Algebra 2

aX(bXc) = (ac)b-(ab)c

(a)
L.H.S. = (ac)b-(ab)c+(ba)c-(bc)a
= (ac)b-(ab)c+(ab)c-(bc)a
= (ac)b-(bc)a
= (ca)b-(cb)a
= cX(bXa)
= -cX(aXb)
= (aXb)Xc
= R.H.S.


(b)
R.H.S. = (ac)b-(ab)c
= mb+nc
This is a vector lies on the plane determined by b and c.

L.H.S. = (aXb)Xc
= -cX(aXb)
= cX(bXa)
= (ca)b-(cb)a
= mb+pa
This is a vector lies on the plane determined by b and a.

The two vectors are not equal unless nc = pa,
-(ab)c = -(cb)a
c = [(cb)/(ab)]a