How do you perform vector arithmetic and other vector functions in pgvector?

-- Vector arithmetic operators:
-- + : element-wise addition
-- - : element-wise subtraction
-- * : element-wise multiplication

SELECT '[1,2,3]'::vector + '[4,5,6]'::vector;  -- [5,7,9]
SELECT '[4,5,6]'::vector - '[1,2,3]'::vector;  -- [3,3,3]
SELECT '[1,2,3]'::vector * '[2,2,2]'::vector;  -- [2,4,6]

-- Compute the L2 norm (magnitude) of a vector:
SELECT l2_norm('[1,2,2]'::vector);  -- sqrt(1+4+4) = 3.0

-- Normalise a vector (make unit length):
SELECT l2_normalize('[1,2,2]'::vector);
-- Result: [0.333, 0.667, 0.667] (each / 3.0)

-- Compute the average (centroid) of multiple vectors:
SELECT avg(embedding) FROM documents WHERE category = 'technical';
-- Returns the centroid vector of all 'technical' embeddings
-- Useful for finding the 'centre' of a cluster

-- Compute the sum of all vectors:
SELECT sum(embedding) FROM documents;

-- Round-trip: store normalised version of an embedding
UPDATE documents
SET embedding = l2_normalize(embedding)
WHERE id = 1;

-- Convert between types:
SELECT embedding::halfvec(1536) FROM documents WHERE id = 1;
SELECT embedding::vector(1536)  FROM documents WHERE id = 1;