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

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SUMMARY

The discussion centers on the validity of taking the square value of a dot product in the context of Compton scattering. It is established that the equation p^2 = p·p holds true, where p represents a vector. The dot product is defined as A·A = A^2 when the vectors are identical, confirming that the operation is mathematically sound. However, the confusion arises when comparing this to the case of distinct vectors, where A·B does not equal ABcosθ unless A and B are the same vector.

PREREQUISITES
  • Understanding of vector mathematics
  • Familiarity with the dot product definition
  • Knowledge of trigonometric functions in relation to vectors
  • Basic principles of Compton scattering
NEXT STEPS
  • Study the properties of the dot product in vector algebra
  • Explore the geometric interpretation of the dot product
  • Learn about vector projections and their applications
  • Investigate the implications of the dot product in physics, particularly in scattering theories
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Students of physics, mathematicians, and anyone interested in advanced vector calculus and its applications in physical theories such as Compton scattering.

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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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