CODING CHALLENGE · N°05

Cosine Similarity

Easy AI EngineeringEmbeddingsMath

The measure behind every embedding search and RAG system: how aligned are two vectors, regardless of their length? It is the dot product divided by both magnitudes — 1 means identical direction, 0 orthogonal, -1 opposite.

The problem

Given two equal-length vectors a and b, return their cosine similarity: dot(a, b) / (‖a‖ · ‖b‖), where ‖v‖ is the Euclidean norm (square root of the sum of squares). The result lies in [-1, 1] and ignores vector length — only direction matters.

EXAMPLE 1

Input a = [1, 2, 3], b = [1, 2, 3]

Output 1.0

same direction

EXAMPLE 2

Input a = [1, 0], b = [0, 1]

Output 0.0

orthogonal

EXAMPLE 3

Input a = [1, 0], b = [-1, 0]

Output -1.0

opposite

EXAMPLE 4

Input a = [1, 1], b = [2, 2]

Output 1.0

length does not matter

CONSTRAINTS

len(a) == len(b) ≥ 1, and neither vector is all zeros.
Result in [-1, 1], within floating-point tolerance.
Cosine ignores magnitude — scaling a vector does not change the answer.

SOLVE IT YOURSELF

Your turn — write it

Edit the stub, hit Run (or ⌘/Ctrl + Enter), and watch the hidden tests. Stuck? the hints are right above and Reveal solution is one click away.

YOUR TASK

Implement cosine_similarity(a, b) = dot product of a and b, divided by the product of their Euclidean norms.

HINTS — 4 IDEAS

The dot product is the sum of element-wise products: Σ aᵢ·bᵢ.
The Euclidean norm of v is sqrt(Σ vᵢ²).
Divide the dot product by ‖a‖ · ‖b‖ to cancel out length and leave only direction.
Identical vectors give 1; orthogonal give 0; opposite give -1.

CPython · WebAssembly

Approach, complexity & discussion — open after you solve

The approach

Compute the dot product of the two vectors, then divide by the product of their magnitudes: cos = (a·b) / (‖a‖‖b‖). Accumulate the dot product and both sums-of-squares in a single pass, take the square roots at the end, and guard against a zero-magnitude vector before dividing.

Complexity

Time O(d) in the dimension; space O(1) — a few running scalars, one pass.

Common mistakes

Returning the raw dot product without dividing by the magnitudes — then length dominates and it is no longer cosine.
Dividing by zero on a zero vector — guard it (define the similarity as 0 or handle explicitly).
Assuming the inputs are the same length without checking.

Where this shows up

Cosine similarity is the standard score for comparing embeddings — semantic search, RAG retrieval, clustering, deduplication, recommendations. On vectors that are already unit-normalized (many embedding models return these) it reduces to a plain dot product, which is exactly why vector databases can rank by dot product and get cosine ordering for free.

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