Vector dot products and cos and Pythagorus

\newcommand{\abs}[1]{\lvert#1\rvert}
\newcommand{\norm}[1]{\lVert#1\rVert}
\abs{\vec{X}}\ \ \ \ \ \   
\norm{\vec{X}}

As it relates to how LaTeX instantiates a new command. It can be seen that the format starts with "newcommand" , the name of the new command, number of arguments, and how it is created using previously defined methods with the #(n) implying an argument pass in curly brackets. It is a lot like a function in shell which uses $1 .... as args passed.

cos\theta = \frac
{\vec{v}\cdot \vec{w}}
{|\vec{v}|\cdot|{\vec{w}|}}

It does seem that the concept of sin and cos are defined in a rather circular way ( no pun intended ). The values at an angle are defined by the relationship of dimension itself and so long as a standard is defined, the numbers are the same in relationship to scale.