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SquaredEuclideanDistance[u,v]

gives the squared Euclidean distance between vectors u and v.

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SquaredEuclideanDistance

SquaredEuclideanDistance[u,v]

gives the squared Euclidean distance between vectors u and v.

Details

Examples

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Basic Examples  (2)

Squared Euclidean distance between two vectors:

Wolfram Language code: SquaredEuclideanDistance[{a, b, c}, {x, y, z}]

Squared Euclidean distance between numeric vectors:

Wolfram Language code: SquaredEuclideanDistance[{1, 2, 3}, {2, 4, 6}]

Scope  (2)

Compute distance between any vectors of equal length:

Wolfram Language code: SquaredEuclideanDistance[RandomReal[5, 100], RandomReal[5, 100]]

Compute distance between vectors of any precision:

Wolfram Language code: SquaredEuclideanDistance[N[{1, 5, 2, 3, 10}, 50], N[{4, 15, 20, 5, 5}, 50]]

Applications  (2)

Cluster data using squared Euclidean distance:

Wolfram Language code: FindClusters[{{2, 3}, {5, 10}, {4, 5}, {2, 2}}, DistanceFunction -> SquaredEuclideanDistance]

Demonstrate the triangle inequality:

Wolfram Language code: d1 = SquaredEuclideanDistance[{a, b}, {a, c}]
Wolfram Language code: d2 = SquaredEuclideanDistance[{a, c}, {d, c}]
Wolfram Language code: d3 = SquaredEuclideanDistance[{a, b}, {d, c}]
Wolfram Language code: Simplify[d3 <= d1 + d2]

Properties & Relations  (4)

SquaredEuclideanDistance is equivalent to the squared Norm of a difference:

Wolfram Language code: SquaredEuclideanDistance[{a, b, c}, {x, y, z}]
Wolfram Language code: Norm[{a, b, c} - {x, y, z}] ^ 2

The square root of SquaredEuclideanDistance is EuclideanDistance:

Wolfram Language code: SquaredEuclideanDistance[{a, b, c}, {x, y, z}]//Sqrt
Wolfram Language code: EuclideanDistance[{a, b, c}, {x, y, z}]

Variance as a SquaredEuclideanDistance from the Mean:

Wolfram Language code: data = {a, b, c}
Wolfram Language code: mean = Table[Mean[data], {Length[data]}]
Wolfram Language code: Variance[data]
Wolfram Language code: % == SquaredEuclideanDistance[data, mean] / (Length[data] - 1)//FullSimplify

SquaredEuclideanDistance computed from RootMeanSquare of a difference:

Wolfram Language code: SquaredEuclideanDistance[{1, 2, 3}, {2, 4, 6}]
Wolfram Language code: Length[{1, 2, 3}]RootMeanSquare[{1, 2, 3} - {2, 4, 6}] ^ 2
Wolfram Research (2007), SquaredEuclideanDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/SquaredEuclideanDistance.html.

Text

Wolfram Research (2007), SquaredEuclideanDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/SquaredEuclideanDistance.html.

CMS

Wolfram Language. 2007. "SquaredEuclideanDistance." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SquaredEuclideanDistance.html.

APA

Wolfram Language. (2007). SquaredEuclideanDistance. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SquaredEuclideanDistance.html

BibTeX

@misc{reference.wolfram_2026_squaredeuclideandistance, author="Wolfram Research", title="{SquaredEuclideanDistance}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/SquaredEuclideanDistance.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_squaredeuclideandistance, organization={Wolfram Research}, title={SquaredEuclideanDistance}, year={2007}, url={https://reference.wolfram.com/language/ref/SquaredEuclideanDistance.html}, note=[Accessed: 13-June-2026]}