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 of dimension 2.

Remarks

Each Euclidean space of dimension 2 is isometric to $\mathbb{R}^2$. A choice of a Cartesian coordinate system 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 $\sqrt{9 + 16} = 5$.