Cosine Similarity

What Cosine Similarity Is

Suppose you have two vectors. Maybe they’re word embeddings: one for “cat” and one for “kitten.” You want to know: how similar are these two meanings?

Cosine similarity measures it by computing the cosine of the angle between the two vectors.

cosine_similarity(A, B) = (A · B) / (|A| × |B|)

Where A · B is the dot product, and |A| is the length (magnitude) of vector A.

The result is always between -1 and +1:

• +1: vectors point in exactly the same direction. Identical meaning (perfect similarity).
• 0: vectors are perpendicular. Unrelated.
• -1: vectors point in exactly opposite directions. Opposite meaning.

The Geometric Intuition

Forget magnitudes for a moment. Think of every vector as an arrow from the origin in some high-dimensional space.

• “Cat” might point northeast.
• “Kitten” might also point northeast, very close to “cat.”
• “Spaceship” points southwest, far from both.

Cosine similarity asks: what’s the angle between two arrows? Small angle = similar. Big angle = different. It ignores how long the arrows are, only their direction.

Why Not Just Use Dot Products?

A dot product is A · B = |A| × |B| × cos(angle). It depends on both the angle AND the magnitudes. Two vectors might have a huge dot product just because they're both very long, even if their angle is wide.

Cosine similarity divides out the magnitudes, leaving only the angle. So it’s direction-only similarity, immune to scale.

This is useful when you don’t want vector magnitude to bias the comparison — like comparing word meanings, where a frequently used word might have a vector with bigger magnitude than a rare word, but you want them compared fairly on meaning.