Strange Dot Product definition

[TonyEsposito]

Hi i have seen in abook the dot product defined as follows:
Dot(A,B)=(1/4)[Norm(A+B)^2-Norm(A-B)^2]
how this definition connect with the common one: Dot(A,B)=Sum(ai*bi)
Thanks!

 

[mathman]

$|A+B|^2-|A-B|^2=|A|^2+2A\cdot B+|B|^2-|A|^2+2A\cdot B-|B|^2=4A\cdot B$.

 

[robphy]

Hi i have seen in abook the dot product defined as follows:
Dot(A,B)=(1/4)[Norm(A+B)^2-Norm(A-B)^2]
how this definition connect with the common one: Dot(A,B)=Sum(ai*bi)
Thanks!

You have written one of the "polarization identities".
You get the "Dot(A,B)=Sum(ai*bi)" formula if your Norm() is the euclidean magnitude of a vector.

 

[TonyEsposito]

Sorry I'm late! I have read the replies some time ago...very useful! Thanks!