# 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 $u→$ and $v→$ with

$u→⋅v→=|u→||v→|cosθ,$

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:

If Then Therefore
$u→⋅v→>0$ $0\text{º}<\theta <90\text{º}$ acute
$u→⋅v→<0$ $90\text{º}<\theta <180\text{º}$ obtuse
$u→⋅v→=0$ $\theta =90\text{º}$ perpendicular

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

$\left[{u}_{1},{u}_{2},{u}_{3}\right]\cdot \left[{v}_{1},{v}_{2},{v}_{3}\right]={u}_{1}{v}_{1}+{u}_{2}{v}_{2}+{u}_{3}{v}_{3}\text{.}$

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:

$a→⋅a→=|a→|2.$

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

$a→⋅b→=b→⋅a→.$

$a→⋅(b→+c→)=a→⋅b→+a→⋅c→.$
$ka→⋅b→=k(a→⋅b→)=a→⋅kb→.$
$a→⋅0→=0.$