How do you use the inner product operator <#> with pgvector and when is it appropriate?

-- <#> returns the NEGATIVE inner product
-- For unit vectors: inner_product = cosine_similarity
-- ORDER BY <#> ASC = most similar first (most negative = highest similarity)

-- Inner product similarity search:
SELECT id, content,
       -(embedding <#> '[0.1,0.2,0.3]') AS inner_product_similarity
FROM documents
ORDER BY embedding <#> '[0.1,0.2,0.3]'  -- ASC order = most similar
LIMIT 5;
-- NOTE: negate in SELECT for display, but use the raw expression in ORDER BY

-- Why negative? PostgreSQL indexes only support ASC scans
-- High inner product = more similar, but ASC would put low values first
-- Solution: return the NEGATIVE so ASC scan gives most similar first

-- When inner product == cosine similarity:
-- When all vectors are unit-normalised (L2 norm = 1.0)
-- OpenAI text-embedding-3-small/large outputs ARE unit-normalised
-- So for OpenAI embeddings, <#> and <=> produce equivalent RANKINGS

-- Verify your vectors are normalised:
SELECT id, l2_norm(embedding) AS norm FROM documents LIMIT 5;
-- Should return values very close to 1.0 for normalised vectors

-- HNSW index for inner product:
CREATE INDEX ON documents USING hnsw (embedding vector_ip_ops);

-- IVFFlat index for inner product:
CREATE INDEX ON documents USING ivfflat (embedding vector_ip_ops) WITH (lists = 100);