Cosine Similarity

EEC 351 · Fundamentals of AI/ML · Prof. Parikshit Pareek & Prof. Jitin Singla · IIT Roorkee

The inner product factorises as x^⊤y = ‖x‖₂ ‖y‖₂ cos θ. Solving for cos θ gives the cosine similarity — a number in [−1, 1] that measures how much two vectors point in the same direction, ignoring their lengths. Drag either arrowhead to change the vectors.

cos θ = x^⊤y / (‖x‖₂ · ‖y‖₂) ∈ [−1, 1]

━━ vector x ━━ vector y

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x = (—) ‖x‖₂ = —

y = (—) ‖y‖₂ = —

x^⊤y = —

θ = —

cos θ = —

−1 (opposite)0 (orthogonal)+1 (aligned)

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Interpretation

Where this appears in ML

Word/sentence embeddings: the cosine between two embedding vectors measures semantic similarity. "king" and "queen" have high cosine similarity; "king" and "banana" have low.
Information retrieval: rank documents by cos(query, document) — independent of how long each document is.
Recommender systems: user-item or item-item similarity via cosine.
Cauchy-Schwarz is exactly the statement that this quantity stays in [−1, 1] — geometrically obvious, algebraically non-trivial.
Not a distance: can be negative, doesn't satisfy the triangle inequality. The "cosine distance" 1 − cos θ patches the first issue.