dot product

The dot product is an inner product over a Euclidean vector space. As an operation, it takes as argument two vectors and returns a single real number.

Formally,
the dot product on a Euclidean vector space ${\mathbb{R}}^n$ can be defined using Cartesian coordinates:

where $x_i$ is the $i^{th}$ element of $\overrightarrow{x}$ as an ordered tuple;
or it can be defined geometrically as

where $\left|\overrightarrow{x}\right|$ is the magnitude of $\overrightarrow{x}$, and $\theta$ is the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.