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