03 · Distance Metrics (Cosine, Euclidean, Dot Product)

Every vector search boils down to "how far apart are these two vectors?" — but "far apart" can mean different things. This lesson covers the three metrics you'll see in every vector database you touch.

Euclidean distance (L2)

Straight-line distance between two points, generalized to N dimensions:

import numpy as np

def euclidean_distance(a, b):
    return np.linalg.norm(a - b)

a = np.array([1.0, 2.0, 3.0])
b = np.array([2.0, 1.0, 4.0])
print(euclidean_distance(a, b))   # sqrt(1+1+1) ≈ 1.732

Euclidean distance cares about absolute position, including magnitude. Two vectors pointing in the exact same direction but with different lengths will have nonzero Euclidean distance between them.

Dot product

def dot_product(a, b):
    return np.dot(a, b)

print(dot_product(a, b))   # 1*2 + 2*1 + 3*4 = 16

The dot product gets larger as vectors point in more similar directions and have larger magnitudes — it's a similarity measure, not a distance (bigger = more similar, unlike the distances above where smaller = more similar).

Cosine similarity

Cosine similarity measures the angle between two vectors, ignoring their magnitude entirely:

def cosine_similarity(a, b):
    return np.dot(a, b) / (np.linalg.norm(a) * np.linalg.norm(b))

a = np.array([1.0, 2.0, 3.0])
b = np.array([2.0, 4.0, 6.0])   # same direction as a, just scaled 2x
print(cosine_similarity(a, b))   # 1.0 — identical direction, "perfectly similar"
print(euclidean_distance(a, b))  # 3.74 — but far apart in Euclidean terms!

This is why cosine is the default for text embeddings: two embeddings for "I love this product" and "I LOVE THIS PRODUCT!!!" might differ in magnitude (different word counts, different emphasis) but should still be judged as pointing in nearly the same semantic direction.

Cosine similarity ranges from -1 (opposite) through 0 (unrelated/ orthogonal) to 1 (identical direction).

When to use which

Metric	Use when...
Cosine	Default for text embeddings — direction (meaning) matters, magnitude doesn't
Euclidean	Absolute position matters — e.g. spatial coordinates, some image embeddings
Dot product	You've already normalized every vector to unit length — then it equals cosine, but is cheaper to compute

Every vector database you'll use in this course (Chroma, Pinecone, Weaviate, Qdrant, pgvector) lets you pick a metric per collection/index — usually cosine, l2 (Euclidean), or dotproduct/ip (inner product).

How It Actually Works

Why does normalizing a vector make dot product behave exactly like cosine similarity?

1. Cosine similarity's formula is (a · b) / (‖a‖ ‖b‖). It's the dot product, scaled down by both vectors' magnitudes.
2. If both vectors are already unit length (‖a‖ = 1 and ‖b‖ = 1), the denominator becomes 1 * 1 = 1, so the formula collapses to just a · b — the raw dot product.
3. This means: normalize once, compare cheaply forever. A production vector database can normalize every embedding exactly once at insert time, then use the plain dot product (a single fused multiply-add loop, extremely cheap on modern CPUs/GPUs with SIMD instructions) for every future comparison — instead of recomputing two norm() calls on every single query-to-vector comparison.
4. This is why most embedding models (including sentence-transformers models you'll use in Lesson 5) ship with a normalize_embeddings=True option — so you can store unit vectors and let the database use the fastest possible similarity operation while still getting cosine semantics.
5. Verify it yourself:

def normalize(v):
    return v / np.linalg.norm(v)

a = np.array([1.0, 2.0, 3.0])
b = np.array([2.0, 1.0, 4.0])

cos_sim = cosine_similarity(a, b)
dot_of_normalized = np.dot(normalize(a), normalize(b))

print(cos_sim, dot_of_normalized)   # same value, two different routes

Exercise

Using the 4 toy documents from Lesson 2 (documents dict with [animals, technology, food, sports] dimensions), write a function that computes the cosine similarity of doc1 against doc2, doc3, and doc4. Print the results sorted from most to least similar, and in a one-line comment explain whether the ranking matches what you'd intuitively expect given the documents' themes.