Is taking the square value of a dot product a valid mathematical operation?

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Discussion Overview

The discussion revolves around the validity of taking the square value of a dot product in mathematical operations, particularly in the context of Compton scattering. Participants explore the implications of this operation and its consistency with the definition of the dot product.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions whether taking the square value of a dot product is valid, referencing its use in the proof of Compton scattering.
  • Another participant suggests using the definition of the dot product to analyze the situation for general vectors.
  • A different participant provides a mathematical expression showing that when two identical vectors are involved, the dot product simplifies to the square of the magnitude of the vector, indicating that this operation holds under specific conditions.
  • One participant expresses confusion about the application of the dot product, noting that it seems to only apply in cases where vectors are identical, and raises concerns about its validity when vectors differ.
  • A participant explains that the dot product can be interpreted as the projection of one vector onto another, reinforcing the idea that when the vectors are the same, the dot product equals the square of the vector's magnitude.

Areas of Agreement / Disagreement

Participants express differing views on the validity of squaring a dot product, with some supporting its application under certain conditions while others question its general applicability. The discussion remains unresolved regarding the broader implications of this operation.

Contextual Notes

Limitations include potential misunderstandings of the dot product's definition and its application to non-identical vectors. The discussion does not resolve these ambiguities.

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I was recently going through the proof of Compton scattering and I saw that they took a square value and wrote it as p^2=p(dot)p= etc... Is this true or all squared values?
 
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Use the definition of dot product and see what you get for a general vector.
 
SteamKing said:
Use the definition of dot product and see what you get for a general vector.

<A,A>=AAcos(0)=AA(1)=AA=A^2

So this only applies when numbers are squared. I just find it strange that the dot product only applies to when normal multiplication is the square of a number and not at any other cases for example if A=/=B AB=/=ABcosθ. I guess I just don't understand where the dot product comes from enough to understand why it works like this.
 
A dot B can be thought of a the projection of A onto B. If A and B are the same vector, then A dot B = A^2
 

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