Vectors and the dot product

A vector is a list of numbers, like (3, 4). You can picture it as an arrow from the origin: 3 steps right and 4 steps up. In code it is simply a list.

Adding and scaling

• Add component by component: (3, 4) + (4, 3) = (7, 7). Arrow picture: put one arrow after the other.
• Scale by a number: 2 × (3, 4) = (6, 8), the same direction, twice as long.

Length

By Pythagoras, the length of (3, 4) is √(3² + 4²) = 5. In any number of dimensions, take the square root of the sum of squares.

The dot product

Multiply matching components and add: (3, 4) · (4, 3) = 12 + 12 = 24. The dot product connects to the angle θ between the arrows:

u · v = |u| × |v| × cos θ

so cos θ = (u · v) ÷ (|u| × |v|).

from math import sqrt, acos, degrees

u = [3, 4]
v = [4, 3]
print("u + v =", [a + b for a, b in zip(u, v)])
print("2u =", [2 * a for a in u])
dot = sum(a * b for a, b in zip(u, v))
length_u = sqrt(sum(a * a for a in u))
length_v = sqrt(sum(b * b for b in v))
print("dot =", dot, " |u| =", length_u, " |v| =", length_v)
print("angle =", round(degrees(acos(dot / (length_u * length_v))), 2), "degrees")

• Dot product 0: the vectors are perpendicular.
• Positive: they point roughly the same way. Negative: roughly opposite ways.

Cosine similarity

Search engines and recommendation systems turn documents, songs or users into long vectors and compare them with cos θ: near 1 means "very similar", near 0 means "unrelated".

Common mistakes

• Multiplying two vectors component by component and calling it the dot product. The dot product is one number: you must add the products.

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