Also known as: L2 norm, Euclidean norm, vector magnitude, vector length
The 2-norm of a vector is its Euclidean length — the square root of the sum of squared components. Normalizing a vector to 2-norm = 1 makes it a unit vector.
The 2-norm of a vector
Geometrically, it’s the straight-line distance from the origin to the point the vector represents. The “2” is because each component is raised to the second power; there are also 1-norms (sum of absolute values) and ∞-norms (max absolute value). In retrieval, “norm” means 2-norm unless stated otherwise.
Most embedding models output vectors with varying magnitudes. To compare two embeddings via cosine similarity , you have to divide by both magnitudes — that’s the
A vector with
v_unit = v / numpy.linalg.norm(v)After normalization, every embedding lives on the surface of a unit hypersphere. Distance and angle become equivalent, and direction is the only signal that matters.
Most production embedding pipelines unit-normalize and lose magnitude information. But some training schemes deliberately encode signal in magnitude — for example, attaching a confidence-like quantity to longer or more salient documents. If you’re using a model that does this, normalizing destroys that signal. Check the model’s documentation before normalizing.
Picture a vector
That has two consequences. First, distance and angle become equivalent — Euclidean distance squared between unit vectors is
The cost is permanent: once you normalize, the original magnitude is gone. If your pipeline stores only normalized vectors and you later realize you wanted the magnitude (for length-based filtering, for confidence weighting, for re-ranking by salience), you have to re-encode the corpus.
It collapses cosine similarity into a plain dot product — no division needed at query time. That's the inner loop of every dense retrieval system, so saving the divide per pair compounds fast over billions of comparisons.
When the model deliberately encodes signal in magnitude — for instance, training schemes that map document salience or confidence to vector length. Normalizing those embeddings throws away signal you paid to put in. Always check the model card.
Unit-normalized vectors quantize more cleanly because all components live in roughly the same range. int8 and even 1-bit quantization recover most of the recall on a normalized index, which is why quantization recipes assume normalized inputs.
