Warning B.5.1. Cosine similarity is not distance.
Cosine similarity compares direction after normalization. It is not the same as Euclidean distance.
Section B.5 Dot Products, Norms, Distances, and Cosine Similarity
This section helps you read short NumPy expressions for dot products, lengths, distances, and direction-based comparisons.
u = np.array([3.0, 4.0])
v = np.array([4.0, 3.0])
dot = u @ v
norm_u = np.linalg.norm(u)
distance = np.linalg.norm(u - v)
cosine = (u @ v) / (np.linalg.norm(u) * np.linalg.norm(v))
Dot product.
u @ v
Read as. The dot product \(u\cdot v\text{.}\)
Shape/return. A scalar.
Used for. Alignment or score.
Norm.
np.linalg.norm(u)
Read as. The norm \(\|u\|\text{.}\)
Shape/return. A scalar.
Used for. Length.
Row-wise norms.
np.linalg.norm(X, axis=1)
Read as. One norm for each row of
X.
Shape/return. A one-dimensional array with one entry per row.
Used for. Row-wise cosine similarities, such as document-ranking scores.
Euclidean distance.
np.linalg.norm(u - v)
Shape/return. A scalar.
Used for. Distance between vectors.
Cosine similarity.
(u @ v) / (...)
Read as. The cosine similarity \(\frac{u\cdot v}{\|u\|\|v\|}\text{.}\)
Shape/return. A scalar.
Used for. Direction-based comparison.
