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

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Taking the square value of a dot product is valid when the vectors are the same, as demonstrated by the equation A dot A equating to A^2. The dot product is defined as the product of the magnitudes of the vectors and the cosine of the angle between them, which simplifies to the square of the vector's magnitude when the vectors are identical. However, this relationship does not hold for distinct vectors, where A dot B does not equal ABcosθ. The confusion arises from the specific conditions under which the dot product behaves like normal multiplication. Understanding the geometric interpretation of the dot product as a projection helps clarify its application in various scenarios.
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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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