TITLE: Is it possible to use Feynman diagrams to represent a dot product $a \cdot b$?
QUESTION [5 upvotes]: Feynman diagrams are topological entities, but they describe linear
  operators
It has been observed that Feynman diagrams are in particular string diagrams (morphisms in monoidal categories)) in a given category of representations 

I want to know if exist a way to represent or explain a dot product $a \cdot b$ using Feynman diagrams

An inner product space ("scalar product") is a vector space $V$ equipped with a (conjugate)-symmetric bilinear form or sesquilinear form: a linear map from the tensor product $V \otimes V$ of $V$ with itself, or of $V$ with its dual module $\bar{V} \otimes V$  to the ground ring $k$.

REPLY [6 votes]: In A Not-so-Characteristic Equation: the Art of Linear Algebra by
Elisha Peterson, the dot product, cross product, and many other linear algebraic operations are described in terms of diagrams.  There are also many good references in the paper for further exploration.
See also this MO post: How can I learn about doing linear algebra with trace diagrams?