# Dot product

This online calculator calculates the dot product of two vectors

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

#### Second vector

Digits after the decimal point: 2
Dot product

### 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
ai is the i-th coordinate,
n is the dimension of the vector space

The geometric definition and algebraic definition are equivalent, while the latter is very simple to calculate.

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