---
tag: 05HQYM0R4G
kind: definition
name: Dot product
msc: [15A63]
defines: 0S52TNNTH9
uses: [0XM8BZD76G]
sources:
  - url: https://en.wikipedia.org/wiki/Dot_product
    locator: Coordinate definition
---

## Definition

Let $n \ge 1$ and let $x = (x_1, \dots, x_n)$ and $y = (y_1, \dots, y_n)$ be vectors of $\R^n$.
The *dot product* of $x$ and $y$ is the real number
$$x \cdot y = \sum_{i=1}^{n} x_i y_i = x_1 y_1 + x_2 y_2 + \cdots + x_n y_n.$$

## Remarks

In matrix notation, $x \cdot y = x^T y$ for column vectors $x$ and $y$.
It is symmetric, bilinear and positive definite, so it is an [[inner product|0NB0AWGJBW]] on $\R^n$.
Geometrically, $x \cdot y = \norm{x} \norm{y} \cos\theta$, where $\theta$ is the angle between $x$ and $y$.
The same formula defines a symmetric bilinear form on $F^n$ for every field $F$.

## Explanations

### L1 human

The dot product multiplies two vectors coordinate by coordinate and adds the results.
For example, $(1, 2, 3) \cdot (4, 5, 6) = 4 + 10 + 18 = 32$.
It is zero exactly when the two vectors are perpendicular, or when one of them is zero.