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.