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# Dot product

This online calculator calculates the dot product of two vectors

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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.

#### 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 latter is very simple to calculate.

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