Symbol u
u is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.
Read this term in its guide →Published equation contexts
Why this formula appears here
The conventional similarity metric is the cosine of the angle between two vectors: . Cosine became conventional for reasons that are mostly good. It is scale-invariant, which matters when vector norms correlate with nuisance properties like token frequency or document length. It reduces to an inner product on normalised vectors, which is cheap and which most approximate indexes support natively. And it is what several influential embedding models were explicitly trained to make meaningful: Sentence-BERT fine-tuned siamese networks precisely so that sentence embeddings could be compared with cosine similarity, cutting a pairwise-comparison workload from roughly 65 hours to…
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Symbol v
v is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.
Read this term in its guide →Numerator: langle u, v rangle
The complete quantity above the fraction bar.
Read this term in its guide →Denominator: lVert u rVert lVert v rVert
The complete quantity below the fraction bar; it must be nonzero for this division.
Read this term in its guide →How to interpret it
With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.
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Published contexts (1)
A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.
Equation 1 · AI Agents & Systems
Embeddings and the Geometry of Similarity
This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.
The conventional similarity metric is the cosine of the angle between two vectors: . Cosine became conventional for reasons that are mostly good. It is scale-invariant, which matters when vector norms correlate with nuisance properties like token frequency or document length. It reduces to an inner product on normalised vectors, which is cheap and which most approximate indexes support natively. And it is what several influential embedding models were explicitly trained to make meaningful: Sentence-BERT fine-tuned siamese networks precisely so that sentence embeddings could be compared with cosine similarity, cutting a pairwise-comparison workload from roughly 65 hours to…
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