Dot Product & Inner Product

Linear Algebra for Machine Learning · Axiom Academy

Measuring similarity and angle between vectors The dot product combines two vectors into a single number by multiplying corresponding components and summing. The result is a single scalar (a number), not a vector. Example: u = [1, 2, 3] and v = [4, 5, 6] u · v = (1)(4) + (2)(5) + (3)(6) = 4 + 10 + 18 = 32 The dot product reveals the angle between two vectors: Where θ is the angle between u and v, and ||u|| and ||v|| are the magnitudes (lengths). u · v > 0: vectors point in similar directions (angle < 90°) u · v = 0: vectors are perpendicular (orthogonal, angle = 90°) u · v < 0: vectors point in opposite directions (angle > 90°) Similarity Measurement: In machine learning, dot product measures how similar two data vectors are. Higher dot product = more similar. Projections: The dot product lets us project one vector onto another, useful in signal processing and statistics. Physics: Work = force · displacement (both vectors), calculated as a dot product. Neural Networks: Computing neuron outputs involves dot products of input vectors with weight vectors. The dot product is one of the most fundamental operations in machine learning, appearing in almost every algorithm.

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