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Euclidean Metric -- from Wolfram MathWorld

  • ️Weisstein, Eric W.

The Euclidean metric is the function d:R^n×R^n->R that assigns to any two vectors in Euclidean n-space x=(x_1,...,x_n) and y=(y_1,...,y_n) the number

d(x,y)=sqrt((x_1-y_1)^2+...+(x_n-y_n)^2),

(1)

and so gives the "standard" distance between any two vectors in R^n.

The Euclidean metric in Euclidean three-space R^3 is given by

(g)_(alphabeta)=[1 0 0; 0 1 0; 0 0 1],

(2)

giving the line element

where Einstein summation has been used.

See also

Distance, Euclidean Space, Minkowski Metric, Vector Norm

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Cite this as:

Weisstein, Eric W. "Euclidean Metric." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EuclideanMetric.html

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