Question

Show that for any two vectors a and b
(a.b)² – [ (a × b ) × b ].a = a²b²

Answer 1

$(a\sdot b)^2-[(a\times b)\times b]\sdot a=(a\sdot b)^2-[b (a\sdot b)-a(b\sdot b)]\sdot a =(a\sdot b)^2-[(a\sdot b)(a\sdot b)-(a\sdot a)(b\sdot b)]=(a\sdot b)^2-[(a\sdot b)^2-a^2b^2]=a^2b^2$