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CosineDistance[u,v]

gives the angular cosine distance between vectors u and v.

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CosineDistance

CosineDistance[u,v]

gives the angular cosine distance between vectors u and v.

Details

Examples

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Basic Examples  (2)

Cosine distance between two vectors:

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

Cosine distance between numeric vectors:

Wolfram Language code: CosineDistance[{1, 2, 3}, {3, 5, 7}]

Scope  (2)

Compute distance between any vectors of equal length:

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

Compute distance between vectors of any precision:

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

Applications  (1)

Cluster data using cosine distance:

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

Properties & Relations  (3)

Cosine distance includes a dot product scaled by norms:

Wolfram Language code: CosineDistance[{a, b, c}, {x, y, z}]
Wolfram Language code: 1 - {a, b, c}.Conjugate[{x, y, z}] / (Norm[{a, b, c}]Norm[{x, y, z}])

Cosine distance includes a dot product scaled by Euclidean distances from the origin:

Wolfram Language code: u = {a, b, c}; v = {x, y, z};
Wolfram Language code: scale = (EuclideanDistance[u, {0, 0, 0}]EuclideanDistance[v, {0, 0, 0}])
Wolfram Language code: CosineDistance[u, v] == 1 - u.Conjugate[v] / scale

CosineDistance of vectors shifted by their means is equivalent to CorrelationDistance:

Wolfram Language code: u = {a, b, c}; v = {x, y, z};
Wolfram Language code: CosineDistance[u - Mean[u], v - Mean[v]] == CorrelationDistance[u, v]
Wolfram Research (2007), CosineDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/CosineDistance.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_cosinedistance, author="Wolfram Research", title="{CosineDistance}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/CosineDistance.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_cosinedistance, organization={Wolfram Research}, title={CosineDistance}, year={2007}, url={https://reference.wolfram.com/language/ref/CosineDistance.html}, note=[Accessed: 12-June-2026]}