Dot Product

A dot product is an algebraic operation that takes two equal length list of numbers, usually two vectors that results in a single number.

it is defined as the sum of the products of numbers of both lists at the same index, or formally:

$$a\cdot b=\sum _{i=1}^{n}ai\cdot bi=a_1{b}_1+a_2{b}_2+\dots a_n{b}_n$$

The dot product is commutative, meaning:

$$\vec{a}\cdot \vec{b}=\vec{b}\cdot \vec{a}$$

And also satisfies distribution so:

$$\vec{a}\cdot (\vec{b}+\vec{c})=\vec{a}\cdot \vec{b}+\vec{a}\cdot \vec{c}$$

Properties

A vector is perpendicular to another if their dot product is 0. In other words:

$$\vec{a}\cdot \vec{b}=0\leftrightarrow \vec{a}\perp \vec{b}$$