Prove that (a-b).{(b-c)x(c-a)}=0 where a,b and c are vectors.

The given expression can be written as ,

(a-b) .{ bxc - bxa - cxc +cxa }

 

= (a-b).{ bxc - bxa - 0 + cxa }

= { a.bxc - a.(bxa) +a.(cxa) -b.(bxc) - b.(bxa) - b.(cxa) }

 

= { [a b c] - 0 + 0 - 0 - 0 - [b c a] }

 

= [a b c] - [a b c] = 0