Careeroza

Measuring Similarity

Three Common Measures

MeasureFormulaNotes
Dot productΣ aᵢ·bᵢFast; affected by vector length
Cosine similaritya·b / (|a|·|b|)Only the angle matters; from -1 to 1. The usual choice for text
Euclidean distance√Σ (aᵢ-bᵢ)²Smaller = more similar

If vectors are normalized (length 1), dot product and cosine give the same ranking, and Euclidean distance ranks the same way too. Many embedding APIs return normalized vectors.

Length Changes the Dot Product, Not the Cosine

Example

Vector b is just a made 3× longer (the same direction, like the same topic said more strongly):

import math

def dot(a, b): return sum(x * y for x, y in zip(a, b))
def norm(a): return math.sqrt(dot(a, a))
def cosine(a, b): return dot(a, b) / (norm(a) * norm(b))
def euclid(a, b): return math.sqrt(sum((x - y) ** 2 for x, y in zip(a, b)))

a = [0.6, 0.8, 0.0]
b = [1.8, 2.4, 0.0]        # same direction, 3x longer
c = [0.0, 0.6, 0.8]        # different direction

for name, v in [("b", b), ("c", c)]:
    print(f"a vs {name}: dot={dot(a, v):.2f}  cosine={cosine(a, v):.2f}  euclid={euclid(a, v):.2f}")

Output

a vs b: dot=3.00  cosine=1.00  euclid=2.00
a vs c: dot=0.48  cosine=0.48  euclid=1.02