Dot Product of Vectors

The dot product is the building block of all matrix multiplication: every cell of A @ B is one dot product. On its own, it turns two lists of numbers into a single number — a total, a weighted score, or a measure of how similar two players are.

Definition

The dot product of two vectors a and b of the same length n is the single number a · b = a[0]*b[0] + a[1]*b[1] + ... + a[n-1]*b[n-1] — multiply matching positions, then add everything up.

• Vector — a 1D array, such as one player's stats.
• Same length — every value in a must have a partner in b.
• Single number (scalar) — unlike a * b, the result is not an array.
• Geometric meaning — a · b = |a| × |b| × cos(θ), where θ is the angle between the vectors. If the dot product is 0, the vectors are perpendicular (orthogonal) — they share nothing in common.

Syntax

np.dot(a, b)          # classic
a @ b                 # modern operator
np.sum(a * b)         # the definition, written out
np.vdot(a, b)         # flattens inputs first (rarely needed)

Example — three ways, one number

import numpy as np
runs    = np.array([72, 12, 45])
weights = np.array([1, 25, 10])
print(np.dot(runs, weights))
print(runs @ weights)
print(np.sum(runs * weights))
print(72*1 + 12*25 + 45*10)

822
822
822
822

How it works: 72×1 + 12×25 + 45×10 = 72 + 300 + 450 = 822. Element-wise runs * weights would give the array [72 300 450]; the dot product adds it up.

Example — a vector dotted with itself, and outer product

v = np.array([1, 2, 3])
print(v @ v, np.dot(v, v))
print(np.outer(v, v))

14 14
[[1 2 3]
 [2 4 6]
 [3 6 9]]

v · v = 1 + 4 + 9 = 14 — the sum of squares. Its square root is the vector's length (the norm, next-but-one topic). np.outer is a different thing: every combination of products, returned as a matrix.

Example — orthogonal vectors and similarity

a = np.array([1, 0])
b = np.array([0, 1])
c = np.array([2, 0])
print(np.dot(a, b), np.dot(a, c))
u = np.array([3, 4]); w = np.array([4, 3])
cos = np.dot(u, w) / (np.linalg.norm(u) * np.linalg.norm(w))
print(round(float(cos), 3))

0 2
0.96

a and b point in perpendicular directions → dot product 0. a and c point the same way → positive. Dividing the dot product by both lengths gives the cosine similarity (from −1 to 1); 0.96 means u and w point in almost the same direction. Recommendation systems and search engines use exactly this to find "similar" items.

Comparison — a * b vs a @ b vs np.outer(a, b)

Common mistakes

• ❌ Different lengths: np.dot([1, 2, 3], [1, 2]) → ValueError: shapes (3,) and (2,) not aligned: 3 (dim 0) != 2 (dim 0). ✅ Vectors must be the same length.
• ❌ Expecting a * b to return a single number. ✅ Use a @ b or np.dot(a, b).
• ❌ Transposing a 1D array to make a "column" (a.T does nothing for 1D). ✅ 1D vectors need no transposing for a dot product; use a.reshape(-1, 1) if you truly need a column.

Interview tip

Q: What does a dot product of zero mean? A: The two vectors are orthogonal (perpendicular). In data terms the features have no linear overlap — e.g. cosine similarity 0 means the two items are unrelated.