Online calculator: Dot product

The calculator below calculates the dot product of two vectors from vector coordinates using algebraic definition. The dot product definitions, both geometric and algebraic, can be found below the calculator.

Dot product

First vector

Second vector

Calculation precision

Digits after the decimal point: 2

Dot product

The dot product or scalar product of two vectors a and b is defined by
$\mathbf {a} \cdot \mathbf {b} =\|\mathbf {a} \|\ \|\mathbf {b} \|\cos(\theta ),$
where
$\|\mathbf {a} \|\$ denotes the magnitude of a vector a,
$\theta$ is the angle between a and b

The name "dot product" is derived from the centered dot " · " that is often used to designate this operation; the alternative name "scalar product" emphasizes that the result is a scalar, rather than a vector, as is the case for the vector product in three-dimensional space.¹

The dot product may also be defined algebraically as
$\mathbf {\color {red}a} \cdot \mathbf {\color {blue}b} =\sum _{i=1}^{n}{\color {red}a}_{i}{\color {blue}b}_{i}={\color {red}a}_{1}{\color {blue}b}_{1}+{\color {red}a}_{2}{\color {blue}b}_{2}+\cdots +{\color {red}a}_{n}{\color {blue}b}_{n},$
where
a_i is the i-th coordinate,
n is the dimension of the vector space