Cosine Similarity

Cosine similarity is a measure of how similar two vectors are, calculated as their dot product divided by the product of their lengths. It depends on direction, not magnitude: vectors pointing the same way score 1, perpendicular vectors score 0 and opposite vectors score -1.

Why it matters

It is the standard way to compare embeddings in semantic search, recommendation and retrieval-augmented generation. Because it ignores length, a short and a long document about the same topic can still match. Many vector databases offer it as a distance option, and if vectors are normalised to length 1, cosine similarity equals the dot product, which is faster to compute.

Example

a = [1, 2, 3]    b = [2, 4, 6]    c = [3, 0, -1]
cos(a, b) = (2+8+18) / (3.742 * 7.483) = 1.0   (same direction)
cos(a, c) = (3+0-3) / (3.742 * 3.162) = 0.0   (unrelated)

Common mistakes

• Treating a score as a universal threshold. A "good" value such as 0.8 depends on the embedding model.
• Using it on vectors from different embedding models, whose spaces are not comparable.
• Confusing similarity with distance: cosine distance is usually 1 minus the similarity.

Related

• Embedding
• Vector database
• Semantic search
• RAG

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