basic

Cosine Similarity from Dot Product and Norms Calculator

cos θ = (a · b) / (‖a‖ · ‖b‖).

Compute the cosine similarity between two vectors when only the dot product and the two L2-norms are pre-computed: cos θ = (a · b) / (‖a‖ · ‖b‖). Range: −1 (opposite) → 0 (orthogonal) → +1 (parallel). Heavily used in IR, NLP, and recommender systems.

Daniel Oh By Daniel Oh
Published Last reviewed 1 min read

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How to use this calculator

  1. Fill in the inputs above using the units you already have.
  2. Values update automatically as you type — no submit button needed.
  3. Hover any result row for the underlying formula and intermediate values.

Formula

cos θ = (a · b) / (‖a‖ · ‖b‖).

In depth

Compute the cosine similarity between two vectors when only the dot product and the two L2-norms are pre-computed: cos θ = (a · b) / (‖a‖ · ‖b‖). Range: −1 (opposite) → 0 (orthogonal) → +1 (parallel). Heavily used in IR, NLP, and recommender systems.