Editing Euclidean distance (section)

=== Higher dimensions ===
[[File:Euclidean distance 3d 2 cropped.png|thumb|upright=1.2|Deriving the <math>n</math>-dimensional Euclidean distance formula by repeatedly applying the Pythagorean theorem]]

In three dimensions, for points given by their Cartesian coordinates, the distance is

<math display=block>d(p,q)=\sqrt{(p_1-q_1)^2 + (p_2-q_2)^2 + (p_3-q_3)^2}.</math>

In general, for points given by Cartesian coordinates in <math>n</math>-dimensional Euclidean space, the distance is<ref>{{citation|title=Geometry: The Language of Space and Form|series=Facts on File math library|first=John|last=Tabak|publisher=Infobase Publishing|year=2014|isbn=978-0-8160-6876-0|page=150|url=https://books.google.com/books?id=r0HuPiexnYwC&pg=PA150}}</ref>

<math display=block>d(p,q) = \sqrt{(p_1- q_1)^2 + (p_2 - q_2)^2+\cdots+(p_n - q_n)^2}.</math>

The Euclidean distance may also be expressed more compactly in terms of the [[Euclidean norm]] of the [[Euclidean vector]] difference:

<math display=block>d(p,q) = \| p - q \|.</math>