Prove that for any three vectors a, b, and c, [(a+b)(b+c)(c+a)]=2[a b c].

LHS:[(a+b)(b+c)(c+a)]=(a+b).[b+cxc+a]=(a+b).[bxc+bxa+cxc+cxa]=a(bxc)+b(cxa)=[a b c]+[b c a]⇒2[a b c]=RHS
Hence proved.