The two principal definitions of the dot product have always seemed incoherent
to me. After trying (and failing) to research this many times, a math.stackexchange.com
question finally gave an explanation that made sense to me. Here is my
interpretation of it — and trust me, if

The dot product is a fundamental vector operation in mathematics. Graphics
programmers are very familiar with it and will tell you **everything** is a
dot product. There is some truth to that statement: mathematics, and
geometry/trigonometry in particular, is very self-referential (much like
music). Concepts can often be found hidden in surprising ways in what appear
to be unrelated areas.

Today's topic is certainly one of them. I've known the dot product for a long
time, but never understood

To review, the dot (or scalar) product of vectors **a** and **b** is:

The key for the connection is expressing the vectors as angles and the dot
product as the subtraction of those angles: let *α* and
*β* be the angles between the *x* axis (i.e. **i**) and
**a** and **b** respectively. In two dimensions:

All that is left is to apply the cosine subtraction formula: