# Definition:Euclidean Metric/Ordinary Space

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## Definition

Let $\R^3$ be the real vector space representation of ordinary $3$d space.

The **Euclidean metric** on $\R^3$ is defined as:

- $\displaystyle d_2 \left({x, y}\right) := \sqrt{\left({x_1 - y_1}\right)^2 + \left({x_2 - y_2}\right)^2 + \left({x_3 - y_3}\right)^2}$

where $x = \left({x_1, x_2, x_3}\right), y = \left({y_1, y_2, y_3}\right) \in \R^3$.

## Also known as

The **Euclidean metric** is sometimes also referred to as **the usual metric**.

## Also see

- Results about
**the Euclidean metric**can be found here.

## Source of Name

This entry was named for Euclid.

## Historical Note

Euclid himself did not in fact conceive of the Euclidean metric and its associated Euclidean Space and Euclidean norm.

They bear that name because the geometric space which it gives rise to is **Euclidean** in the sense that it is consistent with Euclid's fifth postulate.

## Sources

- 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): $\text{III}$: Pythagoras' Theorem