Cosine similarity calculator and vector search

Type or paste two vectors as lists of numbers, separated by commas or spaces; brackets are ignored, so a vector copied from Python or JSON works as it is. The table shows every common measure with the cosine similarity worked out step by step. Below it, the nearest-neighbour experiment builds an IVF index over random clustered vectors and measures how many of the true ten nearest neighbours each setting finds.

An embedding represents an item, such as a word, a sentence, an image or a user, as a vector of numbers produced by a trained model, arranged so that items with similar meaning get vectors that point in similar directions. Mikolov and colleagues’ word2vec (2013) learned such vectors for words from large amounts of text; today sentence and document embeddings drive semantic search, recommendations and the retrieval step of retrieval-augmented generation.

Using embeddings comes down to two operations: measuring how similar two vectors are, and finding the most similar vectors to a query among millions.

The dot product A·B multiplies matching entries and adds them up. It is large when the vectors point the same way and are long. Cosine similarity divides out the lengths, cos(A, B) = A·B / (|A| |B|), and so depends only on the angle: 1 for the same direction, 0 for perpendicular, −1 for opposite. Euclidean distance measures the straight-line gap between the points.

For vectors scaled to length 1 the three agree: the dot product equals the cosine, and the squared Euclidean distance is 2 − 2·cos, so all three rank neighbours identically. That is why embedding pipelines usually normalise vectors once and then use the cheapest measure. Without normalising they can disagree sharply: (1, 2, 3) and (10, 20, 30.5) have a cosine similarity of 0.99997 yet are 34.08 apart, while (1, 2, 3) and (2, 4, 6) have a cosine of exactly 1 and a distance of 3.74.

Embeddings have hundreds or thousands of dimensions, and high-dimensional space behaves strangely: the distances from a point to random others bunch together. For 1,000 random points in a unit cube, the relative gap between the farthest and nearest point from a random query, (farthest − nearest) / nearest, averaged over 20 queries:

Finding the exact nearest neighbours means comparing the query with every vector, which is too slow for millions of them. Approximate methods look at a small part of the collection and accept missing a few true neighbours. An inverted file (IVF) index, one of the structures in the Faiss library, runs k-means to split the vectors into nlist clusters; a query is compared with the cluster centres and then only with the vectors in the nprobe closest clusters. On 5,000 vectors in 32 dimensions with 64 lists: