---
tag: 0XW7V9ZZ05
kind: object
name: Euclidean plane
aliases: [cartesian-plane]
msc: [51M05]
instance_of: [09XNAPTJQ7]
sources:
  - url: https://en.wikipedia.org/wiki/Euclidean_plane
    locator: lead section
same_as:
  wikidata: Q17285
  wikipedia: Euclidean plane
---

## Description

The Euclidean plane is the set $\mathbb{R}^2$ of the ordered pairs $(x, y)$ of real numbers, with the dot product $(x_1, y_1) \cdot (x_2, y_2) = x_1 x_2 + y_1 y_2$.
The elements of $\mathbb{R}^2$ are the points of the plane.
The dot product gives the length of a vector, the distance between two points and the angle between two vectors.
The distance between the points $p = (p_1, p_2)$ and $q = (q_1, q_2)$ is $\sqrt{(p_1 - q_1)^2 + (p_2 - q_2)^2}$.
The Euclidean plane is the [[Euclidean space|0XPR1E85W0]] of dimension 2.

## Remarks

Each Euclidean space of dimension 2 is isometric to $\mathbb{R}^2$. A choice of a [[Cartesian coordinate system|0RWKYX8339]] gives such an isometry.
The plane with a fixed Cartesian coordinate system is also called the Cartesian plane.
Books I to IV and VI of Euclid's Elements study the geometry of the plane.

## Explanations

### L1 human

Think of the Euclidean plane as an infinite flat sheet of paper. Two numbers, the coordinates $x$ and $y$, fix the position of each point. For example, the points $(0, 0)$ and $(3, 4)$ are at distance $\sqrt{9 + 16} = 5$.