Cosine Similarity by Hand

What we're building

A function that takes two embeddings from Lesson 2 and returns one number: how similar their meanings are. No library does this for us here, it's about fifteen lines of ordinary math, so there's no mystery left in what "similarity search" actually means.

What "similar" means as a number

Picture each 768-number vector as an arrow pointing from the origin out into a 768-dimensional space (impossible to draw, but the 2D and 3D intuition still holds). Two vectors representing similar meanings tend to point in roughly the same direction, even if one is longer than the other. Cosine similarity measures exactly that: the angle between two vectors, ignoring their length entirely.

The result is always between -1 and 1:

• 1 means the two vectors point in exactly the same direction (as similar as it gets).
• 0 means they're at a right angle (unrelated).
• -1 means they point in opposite directions (opposite meaning), rare in practice for embeddings of ordinary text.

The code, piece by piece

dot_product = sum(x * y for x, y in zip(a, b))

The dot product: pair up each vector's numbers by position (zip), multiply each pair, and add up all 768 results into one number. Two vectors pointing the same way produce a large dot product; pointing in unrelated directions, matching positions are as likely to have opposite signs as the same sign, so the products partly cancel out and the sum stays small.

magnitude_a = math.sqrt(sum(x * x for x in a))

A vector's magnitude is its own length, computed the same way you'd find the length of a 2D arrow with the Pythagorean theorem (sqrt(x² + y²)), just extended to all 768 numbers instead of 2. This matters because a longer vector produces a bigger dot product purely from being longer, not from being more similar, we need to divide that effect back out.

return dot_product / (magnitude_a * magnitude_b)

Dividing the dot product by both magnitudes cancels out each vector's own length, leaving a number that depends only on the angle between them, exactly the cosine of that angle, which is where the name comes from.