Similarity & Distance

Use cosine when only direction matters, the default for text embeddings, where a longer document should not outrank a short one just because its vector is bigger. Use the dot product when magnitude carries signal, for example when a model lets a vector's norm encode confidence or popularity and you want stronger vectors to rank higher. Use L2 when you genuinely care about geometric proximity, such as clustering or coordinates in a learned space.

Cosine vs dot product

The normalization identity

Here is the fact that removes most of the confusion. If every vector is L2-normalized so that $\lVert a\rVert = 1$, the three metrics agree on order:

• Cosine equals the dot product, because the denominator is $1$.
• Squared Euclidean distance becomes $\lVert a-b\rVert^2 = 2 - 2\,(a\cdot b)$, so a larger dot product forces a smaller distance.

Therefore ranking by cosine = ranking by dot product = inverse ranking by squared L2 distance. On normalized data you can use whichever kernel your index computes fastest and get identical neighbors.

Worked example

Work in three dimensions. Take the query $q = (1, 1, 0)$, with $\lVert q\rVert = \sqrt{2} \approx 1.4142$, and compare two candidates.

The dot product ranks A first because $3 > 2$, A wins purely because its vector is longer. Cosine ranks B first because $1.0 > 0.7071$: B points in exactly the query's direction (a perfect, $0°$ match), while A sits $45°$ off. If these were text embeddings, B is the better result, and cosine is the metric that surfaces it; the raw dot product was fooled by magnitude. Normalize all three vectors to unit length and the dot products become $0.7071$ for A and $1.0$ for B, identical to the cosine scores, and now the dot product agrees that B ranks first.

Further reading

• scikit-learn: pairwise metrics and distances
• Sentence-Transformers: semantic search
• pgvector: distance operators (<->, <#>, <=>)
• FAISS: metric types (inner product vs L2)