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How does similarity search work — cosine, dot product and Euclidean?

Q: How does similarity search work — cosine, dot product and Euclidean?

**TL;DR.** **Cosine** compares direction (angle), ignoring magnitude — the default for text embeddings. **Dot product** equals cosine on unit-normalized vectors; **Euclidean** measures straight-line distance. Exact search is O(n), so real systems use **ANN indexes** (HNSW, IVF) trading a little recall for huge speed. doc A query doc B <path class='d-edge' d='M 75 96 A 30 30 0 0 1 82 1

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Understand the problem

The metrics behind nearest-neighbour retrieval and why ANN indexes make it fast.

similaritycosineannhnsw

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pgvector: metric operators and an HNSW index

-- cosine distance operator: <=>   (dot: <#>, euclidean: <->)
CREATE INDEX ON chunks USING hnsw (embedding vector_cosine_ops);

SELECT id, title, 1 - (embedding <=> $1) AS similarity
FROM chunks
WHERE tenant_id = $2            -- filter + ANN together
ORDER BY embedding <=> $1
LIMIT 10;

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