Dot product

Dot product = sum of products of corresponding entries

The dot product of two vectors is calculated by multiplying their corresponding entries and summing the results. This operation results in a scalar value, not a vector. The dot product is essential for determining the angle between vectors and for projecting one vector onto another.

Example

For vectors A = [a1, a2] and B = [b1, b2], the dot product A · B = a1*b1 + a2*b2. If A = [2, 3] and B = [4, 5], then A · B = 2*4 + 3*5 = 8 + 15 = 23.

Understanding the dot product is crucial for applications in physics, engineering, and computer science, where vector operations are fundamental.

Related concepts

Matrix norm

L1 norm of a vector is the sum of absolute values of its components

Normalization (machine learning)

L2 normalization equation: x_i' = x_i / ||x||_2

Cosine similarity

Cosine similarity formula: cos(θ) = (A · B) / (||A|| ||B||)

Rotation matrix

Determinant of a 2x2 matrix: ad - bc

Regression analysis

Linear regression equation: ŷ = β0 + β1X

convolution (f * g)(t) = ∫f(τ)g(t-τ)dτ

(f * g)(t) = ∫f(τ)g(t-τ)dτ

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