Calculate cosine similarity by hand

WHAT YOU WILL BUILD

A similarity calculation that rejects zero-length and incompatible vectors.

Cosine similarity divides the dot product by the two vector lengths. For this nonzero real-vector example, scaling a vector without changing its direction does not change cosine similarity. The number measures a geometric relationship, not factual truth.

FOLLOW ALONG

Work through the example.

1. 01

   Predict three similarities

   Compare [1,0] with [10,0], [0,1], and [-1,0]. Predict 1, 0, and -1 before running the script.

2. 02

   Run the calculation

   Check the three printed values. Follow the dot product and normalization terms in the cosine function, especially why [10,0] has the same direction as [1,0].

3. 03

   Inspect rejected inputs

   The assertions reject a zero vector and unequal dimensions. Try a NaN coordinate as another invalid input. A production implementation must also follow its embedding model's metric contract.

4. 04

   Compare a fourth direction

   Add [1,1]. Its cosine with [1,0] is approximately 0.7071. Use a numeric tolerance in your assertion instead of exact equality for the rounded value.

THE COMPLETE LAB

Run it locally.

Run this command from the folder containing your downloaded file:

node vector-similarity.mjs

// SaveMyToken local lab. Run with Node.js 22 or newer.
import assert from "node:assert/strict";

// Handwritten 2D vectors for arithmetic; not language embeddings.
function cosine(a, b) {
  if (!a.length || a.length !== b.length || ![...a, ...b].every(Number.isFinite))
    throw new Error("Invalid vectors");
  const normA = Math.hypot(...a), normB = Math.hypot(...b);
  if (normA === 0 || normB === 0) throw new Error("Zero vector");
  return a.reduce((sum, x, i) => sum + x * b[i], 0) / normA / normB;
}
assert.throws(() => cosine([0, 0], [1, 0]), /Zero vector/);
assert.throws(() => cosine([1], [1, 0]), /Invalid vectors/);
assert.equal(cosine([1, 0], [10, 0]), 1);
console.log("Same direction:", cosine([1, 0], [10, 0]).toFixed(2));
console.log("Perpendicular:", cosine([1, 0], [0, 1]).toFixed(2));
console.log("Opposite:", cosine([1, 0], [-1, 0]).toFixed(2));

Expected output for the unchanged example

Same direction: 1.00
Perpendicular: 0.00
Opposite: -1.00

The assertions also check the baseline behavior. After changing an input, predict the result and update the relevant assertion.

EXPLAIN WHAT YOU LEARNED

Interview practice

Try answering aloud before opening the reference answer. These are original learning questions, not a record of any employer's interviews.