Vector Operations

Dot product

$a \cdot b = \sum_i a_i b_i$

a @ b or np.dot(a, b)

L1 norm

$\lVert a \rVert_1 = \sum_i |a_i|$

np.linalg.norm(a, 1)

L2 norm

$\lVert a \rVert_2 = \sqrt{\sum_i a_i^2}$

np.linalg.norm(a)

L∞ norm

$\lVert a \rVert_\infty = \max_i |a_i|$

np.linalg.norm(a, np.inf)

Euclidean distance

$\lVert a - b \rVert_2 = \sqrt{\sum_i (a_i - b_i)^2}$

np.linalg.norm(a - b)

Cosine similarity

$\dfrac{a \cdot b}{\lVert a \rVert_2 \, \lVert b \rVert_2}$

a @ b / (np.linalg.norm(a) * np.linalg.norm(b))

Angle between

$\theta = \arccos\!\big(\cos\text{-sim}\big)$

np.arccos(cos_sim)

Projection of $a$ onto $b$

$\dfrac{a \cdot b}{\lVert b \rVert_2^2}\, b$

(a @ b / (b @ b)) * b

Unit / normalized vector

$\hat{a} = a / \lVert a \rVert_2$

a / np.linalg.norm(a)