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here a,b,c are all vector quantities
simplify [ a-b,b-c,c-a]
=( a-b) . ( (b-c) x (c-a) )
= (a-b) . ( b x c - b x a -c x c + c x a )
= (a-b) .( bxc + a x b +c x a)
=a.(b x c) +a.(ax b) + a. (c x a) - b. ( b x c) - b.(a x b) -b.(c x a)
= [ a b c] + 0 + 0 - 0 - 0 - [b c a]
since [ a b c] = [b c a]