Dot product

The dot product, also known as the scalar product or inner product, is an operation that takes two vectors and produces a scalar. Geometrically, we define the dot product of $\overrightarrow{u}$ and $\overrightarrow{v}$ with

$\overrightarrow{u}\cdot \overrightarrow{v}=\left|\overrightarrow{u}\right|\left|\overrightarrow{v}\right|\cos \theta \text{,}$

where $\theta$ is the angle between the vectors in standard position. The dot product has a few useful properties that arise from the cosine term:

Algebraically, we define the dot product in ${ℝ}^3$ with

$[u_1,u_2,u_3]\cdot [v_1,v_2,v_3]=u_1{v}_1+u_2{v}_2+u_3{v}_3\text{.}$

Essentially, we get the product of the first components, of the second components, etc., and then take the sum of the products. It works in the same way with ${ℝ}^2$ and any other Euclidean space.

The dot product has a few properties that are worth committing to memory. First, the dot product of a vector with itself is equivalent to the square of its magnitude:

$\overrightarrow{a}\cdot \overrightarrow{a}={\left|\overrightarrow{a}\right|}^2\text{.}$

The dot product is commutative—order doesn’t matter:

$\overrightarrow{a}\cdot \overrightarrow{b}=\overrightarrow{b}\cdot \overrightarrow{a}\text{.}$

It is distributive over addition:

$\overrightarrow{a}\cdot (\overrightarrow{b}+\overrightarrow{c})=\overrightarrow{a}\cdot \overrightarrow{b}+\overrightarrow{a}\cdot \overrightarrow{c}\text{.}$

It is associative over scalar multiplication:

$k\overrightarrow{a}\cdot \overrightarrow{b}=k(\overrightarrow{a}\cdot \overrightarrow{b})=\overrightarrow{a}\cdot k\overrightarrow{b}\text{.}$

Finally, any dot product with the zero vector produces the scalar zero:

$\overrightarrow{a}\cdot \overrightarrow{0}=0\text{.}$