Euclidean space

The Euclidean space of dimension $n$ refers to the set of points $(x_1, x_2, \ldots, x_n)$, where each $x_i$ is a real number. The two-dimensional Euclidean space is the Cartesian plane, and so forth. In this representation, the study of Euclidean space is called coordinate geometry.

Euclidean space can also be viewed as an example of one of several more general mathematical objects. For example, Euclidean space is a metric space with respect to the distance metric, $d(\bold{x},\bold{y}) = \sqrt{(x_1-y_1)^2 + \cdots + (x_n - y_n)^2}$. Similarly, the Euclidean space of dimension $n$ is the unique (up to isomorphism) vector space of dimension $n$ over $\mathbb{R}$ (under pointwise addition and the "usual" scalar multiplication $c \cdot (x_1, \ldots, x_n) = (c x_1, \ldots, c x_n)$ for $c, x_i \in \mathbb{R}$).

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