The Three Distance Operators

Where we left off

Lesson 4 got real embeddings into a table. This lesson measures how far apart two of them are, which is the entire mechanism behind "search by meaning": find the stored vectors closest to a query vector.

Three ways to measure "closest"

pgvector adds three operators, each a different definition of distance between two vectors:

• <->: L2 (Euclidean) distance, straight-line distance between the two points.
• <#>: negative inner product, more similar means smaller (it's negated so "smaller is closer" still holds).
• <=>: cosine distance, more similar in direction, ignoring vector length.

SELECT embedding <-> %s AS l2_distance FROM notes WHERE id = 1;
SELECT embedding <#> %s AS neg_inner_product FROM notes WHERE id = 1;
SELECT embedding <=> %s AS cosine_distance FROM notes WHERE id = 1;

All three take a stored vector on the left and a query vector on the right (or vice versa, the operators are symmetric), and return a single number.

Why three, and which to use

Gemini's embeddings (like most modern embedding models) are trained so that cosine distance is the intended similarity measure, meaning is carried by the direction the vector points, not by how long it is. <=> is what this course uses for every real search from here on. <-> (L2) is the more general-purpose "distance in space" measure used in other domains (like plain coordinates); <#> (inner product) is fastest to compute but only meaningful when vectors are pre-normalized to the same length, Lesson 13 comes back to exactly this tradeoff.

Seeing it directly

query_vector = embeddings_model.embed_query("recipes and cooking")
row = conn.execute(
    "SELECT content, embedding <=> %s AS distance FROM notes ORDER BY distance LIMIT 1",
    (Vector(query_vector),),
).fetchone()

Notice this is the same shape of query used to explore in Lesson 3, just with the query vector as a parameter instead of comparing two stored rows. Running all three operators against a query with no exact word overlap with any chunk (the same trick LangChain Lesson 28 used) confirms this is matching on meaning, not keywords, no matter which operator is used.