What is cosine similarity?

Cosine similarity: an example

For a = [1, 0] and b = [1, 1], cos = (1) / (1 * 1.414) = 0.707, an angle of 45 degrees.

Cosine similarity in Python

A short example you can run (Python 3.11, NumPy). The output below is what it really prints.

import numpy as np

def cosine(a, b):
    return a @ b / (np.linalg.norm(a) * np.linalg.norm(b))

a = np.array([1, 2, 3])
print(round(cosine(a, np.array([2, 4, 6])), 3))    # same direction
print(round(cosine(a, np.array([3, 0, -1])), 3))   # perpendicular
print(round(cosine(a, -a), 3))                     # opposite

Output

1.0
0.0
-1.0

Related terms

• Dot product
• L2 norm
• Euclidean distance
• Unit vector

More Linear algebra terms

• Covariance matrix
• Cross product
• Dense matrix
• Design matrix
• Determinant
• Diagonal matrix
• Diagonalisation
• Dimension

Where you learn it

Cosine similarity is part of the AI Engineering Program, which teaches it with notes, a hands-on lab and an exam. The first lesson is free.