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Euclidean Distance

In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occasionally being called the Pythagorean distance.

Euclidean distance between two points

Distance formulas

One dimension

The distance between any two points on the real line is the absolute value of the numerical difference of their coordinates

One dimension formula

Two dimensions

Two dimensions formula

Higher dimensions

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

Three dimensions formula

Example: the distance between the two points (8,2,6) and (3,5,7):

3-dimension example

In general, for points given by Cartesian coordinates in n-dimensional Euclidean space, the distance is

n-dimensional formula

References