Euclidean space

Description

Let $n$ be a positive integer. The Euclidean $n$-space is the set $\mathbb{R}^n$ of the $n$-tuples $x = (x_1, \dots, x_n)$ of real numbers, with the dot product $$x \cdot y = x_1 y_1 + x_2 y_2 + \dots + x_n y_n.$$ The elements of $\mathbb{R}^n$ are its points, and also its vectors. The dot product gives the length $\|x\| = \sqrt{x \cdot x}$ of a vector $x$. It also gives the distance $\|x - y\|$ between two points $x$ and $y$, and the angle between two nonzero vectors. For $n = 2$ this is the Euclidean plane, and for $n = 3$ it is the space of elementary solid geometry.