Module 5 · Lesson 1

Vectors & Dot Products

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The dot product of two same-length vectors, np.dot(a, b) or simply a @ b, multiplies corresponding elements and sums the results. It shows up everywhere: it’s the core of matrix multiplication (next lesson), and np.linalg.norm(v) — a vector’s length/magnitude — is just sqrt(v @ v).

A useful derived quantity is cosine similarity, which measures the angle between two vectors regardless of their length:

cos(theta) = (a @ b) / (||a|| * ||b||)

It ranges from -1 (opposite direction) to 1 (same direction), with 0 meaning the vectors are orthogonal.

a = np.array([1.0, 2.0, 3.0])
b = np.array([4.0, 5.0, 6.0])

print("dot:   ", np.dot(a, b), a @ b)
print("norm a:", np.linalg.norm(a), np.sqrt(a @ a))

cos_theta = (a @ b) / (np.linalg.norm(a) * np.linalg.norm(b))
print("cos similarity:", cos_theta)

Output:

dot:    32.0 32.0
norm a: 3.7416573867739413 3.7416573867739413
cos similarity: 0.9746318461970762
1.1

Unit vector

Write to_unit_vector(v) that returns v scaled to have length (norm) exactly 1, pointing in the same direction. Use np.linalg.norm.

💡 Hint

Divide v by np.linalg.norm(v).

✅ Solution
def to_unit_vector(v):
    return v / np.linalg.norm(v)
1.2

Cosine similarity

Write cosine_similarity(a, b) that returns the cosine similarity between two 1-D vectors, as defined above.

💡 Hint

(a @ b) / (np.linalg.norm(a) * np.linalg.norm(b)) — just transcribe the formula above.

✅ Solution
def cosine_similarity(a, b):
    return (a @ b) / (np.linalg.norm(a) * np.linalg.norm(b))