Cosine, Dot Product, and Euclidean Similarity

AI systems turn text, images, and users into embeddings: lists of numbers that capture meaning. To find related items, we need a way to compare those vectors.

Let us use:

• A = [1, 2]
• B = [2, 4] — same direction as A, but larger
• C = [2, 1] — close to A, but points in a different direction

Cosine similarity: compare direction

Cosine similarity ignores magnitude and measures the angle between vectors. Its value usually ranges from -1 to 1; higher means more similar.

• cosine(A, B) = 1.0
• cosine(A, C) = 0.8

Although B is larger, it points in exactly the same direction as A. Cosine similarity is a strong default for semantic search, where meaning matters more than vector size.

Dot product: compare direction and magnitude

The dot product multiplies matching dimensions and adds them. A larger score means stronger alignment, but vector magnitude also affects the result.

• A · B = (1 × 2) + (2 × 4) = 10
• A · C = (1 × 2) + (2 × 1) = 4

Dot product is useful when magnitude carries information, and it is especially fast for ranking normalized embeddings. With normalized vectors, dot product and cosine similarity are equivalent.

Euclidean distance: compare physical closeness

Euclidean distance is the straight-line distance between two points. Here, smaller means more similar.

• distance(A, B) = √5 ≈ 2.24
• distance(A, C) = √2 ≈ 1.41

C is physically closer to A, even though B points in the same direction. Euclidean distance works well when absolute position and scale matter, such as clustering spatial features.

Quick rule of thumb

• Use cosine similarity for direction or semantic meaning.
• Use dot product when direction and magnitude both matter.
• Use Euclidean distance when geometric closeness matters.

The best metric depends on what the embedding's magnitude represents—and whether it should influence similarity at all.