Question

Someone explains me, please
Every vector on C has the form
\(A_{ab}= az + b\bar z\) for some a,b in C
Let u= (u1,u2), v =(v1,v2)
(u, v)=((u1,u2),(v1,v2)) = u1v1+u2v2
I got it. What I didn't get is
(u,v) = \(Re(u,\bar v)=Re(\bar u, v)\)
Give me example, please