Why is cosine similarity preferred over Euclidean distance for text embeddings?

**Cosine similarity** measures the angle between two vectors. **Euclidean distance** measures the straight line distance between their tips. For text embeddings, angle is what carries semantic meaning, and the straight line distance gets polluted by vector magnitude. Text embedding models do not always produce vectors of the same length. Longer documents, denser passages, and slight differences in tokenisation can change the magnitude of the resulting vector even when the meaning is the same. If you use Euclidean distance, two vectors describing the same concept but with different magnitudes will appear far apart because their tips end up in different places in space. Cosine ignores this because two vectors pointing in the same direction always have an angle of zero between them, regardless of how long they are. A useful mental picture: cosine asks where on the unit sphere a vector points. All vectors get projected onto the sphere, and similarity is measured there. Two passages on the same topic should point to nearby spots on the sphere, even if one is a sentence and the other is a paragraph. Many modern embedding APIs normalise vectors to unit length before returning them. When all vectors are unit length, cosine and Euclidean ranking are equivalent, because the only difference between two unit vectors is their angle. But you cannot assume this is happening unless the documentation says so.

**Cosine similarity** wins for text embeddings because direction in embedding space carries semantic meaning, while magnitude carries mostly uninteresting variation like document length and token density. The definitions diverge on the magnitude axis. Cosine similarity is one formula. Euclidean distance is another. $$\text{cos}(u, v) = \frac{u \cdot v}{\lVert u \rVert \, \lVert v \rVert}$$ $$d_{\text{euc}}(u, v) = \lVert u - v \rVert$$ Cosine divides out both norms. Two vectors pointing in the same direction always score 1 regardless of length. Euclidean distance grows with the gap in magnitudes even when the directions match. Since longer passages tend to produce vectors with larger norms in many embedding models, Euclidean conflates length with semantic difference. In practice the distinction collapses when vectors are pre-normalised to unit length, because two unit vectors differ only by angle. Most modern embedding APIs from OpenAI, Voyage, Cohere, and Nomic return unit-norm vectors, which means cosine and Euclidean rank identically. But the conceptual reason cosine is the right default is the magnitude-invariance argument. That argument is what matters when you build the index or pick a similarity metric.