Prove the identity u v 14 ll u v ll2 ll u v ll2Soluti

Prove the identity

< u , v > = (1/4)( ll u + v ll^2 - ll u - v ll^2)

Solution

RHS = 1/4 ll u+v ll^2-1/4 ll u-v ll^2
= 1/4 (u+v).(u+v) -1/4 (u-v).(u-v)
= 1/4 [u.u + u.v + v.u + v.v] - 1/4 [u.u - u.v - v.u + v.v]
= 1/4 [||u||^2 + 2u.v +|| v||^2] - 1/4 [||u||^2 - 2u.v +|| v||^2]
=1/4 [2u.v + 2.v] = 1/4 .4u.v = u.v = LHS

Hence, proved the given identity.