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

[Eats Dirt]

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?

 

[SteamKing]

Use the definition of dot product and see what you get for a general vector.

 

[Eats Dirt]

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.

 

[SteamKing]

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

 

[CellCoree]

I can confirm that the value squared of a dot product is indeed a valid mathematical operation. In the context of Compton scattering, the dot product is used to calculate the momentum transfer between a photon and an electron. Taking the square value of this dot product allows for a simpler and more concise representation of the momentum transfer equation.

In general, taking the square value of a dot product is applicable in cases where the dot product is used to calculate a quantity that is squared, such as momentum or energy. However, it is not always necessary to take the square value, as it depends on the specific equation or problem being solved.

In summary, the use of value squared in a dot product is a valid mathematical operation, and it is commonly used in various scientific fields, including physics and engineering. It is important to understand the context and reasoning behind taking the square value in order to accurately apply it in calculations.