Strange Dot Product definition

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Discussion Overview

The discussion revolves around the definition of the dot product in vector mathematics, specifically comparing a less common definition involving norms to the standard definition based on component-wise multiplication. The scope includes theoretical exploration and mathematical reasoning.

Discussion Character

  • Exploratory, Technical explanation, Mathematical reasoning

Main Points Raised

  • One participant presents a definition of the dot product as Dot(A,B)=(1/4)[Norm(A+B)^2-Norm(A-B)^2] and seeks clarification on its connection to the standard definition Dot(A,B)=Sum(ai*bi).
  • Another participant provides a mathematical derivation showing that the expression simplifies to 4A·B, suggesting a relationship between the two definitions.
  • A third participant notes that the presented definition is one of the "polarization identities" and indicates that the standard definition arises when the norm is interpreted as the Euclidean magnitude of a vector.
  • A later reply expresses appreciation for the responses received, indicating that the information was helpful.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the implications of the definitions, but there is agreement on the mathematical relationship between the two forms of the dot product as discussed.

Contextual Notes

The discussion does not resolve potential limitations or assumptions regarding the definitions of norms or the specific conditions under which the polarization identity holds.

TonyEsposito
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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!
 
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##|A+B|^2-|A-B|^2=|A|^2+2A\cdot B+|B|^2-|A|^2+2A\cdot B-|B|^2=4A\cdot B##.
 
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TonyEsposito said:
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.
 
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Sorry I'm late! I have read the replies some time ago...very useful! Thanks!
 
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