Enter two vectors — get the dot product with full step-by-step breakdown, angle between vectors, and geometric interpretation.
A · B = a₁b₁ + a₂b₂ + ⋯ + aₙbₙ = |A||B| cos θ
For two vectors A = (a₁, a₂, …, aₙ) and B = (b₁, b₂, …, bₙ), the dot product sums the products of corresponding components.
The geometric form |A||B| cos θ reveals the angle between vectors. When A · B = 0, the vectors are orthogonal (perpendicular). Positive means roughly same direction; negative means opposing.
One-on-one linear algebra tutoring — from dot products and cross products through eigenvalues, SVD, and beyond.