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.


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