Math 2.0

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Euclidean space

Description

Let nn be a positive integer. The Euclidean nn-space is the set Rn\mathbb{R}^n of the nn-tuples x=(x1,…,xn)x = (x_1, \dots, x_n) of real numbers, with the dot product x⋅y=x1y1+x2y2+⋯+xnyn.x \cdot y = x_1 y_1 + x_2 y_2 + \dots + x_n y_n. The elements of Rn\mathbb{R}^n are its points, and also its vectors. The dot product gives the length ∥x∥=x⋅x\|x\| = \sqrt{x \cdot x} of a vector xx. It also gives the distance ∥x−y∥\|x - y\| between two points xx and yy, and the angle between two nonzero vectors. For n=2n = 2 this is the Euclidean plane, and for n=3n = 3 it is the space of elementary solid geometry.

Remarks

In a more abstract form, a Euclidean space is a real affine space of finite dimension whose space of translations has an inner product. Each such space of dimension nn is isometric to Rn\mathbb{R}^n with the dot product. The notation En\mathbb{E}^n is also common for the Euclidean nn-space.

Explanations

L1 human

The Euclidean nn-space is the space of ordinary geometry in nn dimensions. A point is a list of nn coordinates. The dot product lets you measure lengths and angles, so the familiar facts of school geometry hold in each dimension.

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  • section Invariants: a table of the main invariants (0 of 1)

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Sources

  1. Web https://en.wikipedia.org/wiki/Euclidean_space, Prototypical examples

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