Cosine Similarity
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)
Quick presets
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.