Classical Mechanics — 2.6: The Cross Product

Full course — free exercises, Feynman reviews, and AI-graded feedback: https://ludium.ai/courses/classical-m... The cross product takes two vectors in a plane and produces a new vector pointing perpendicular to that plane. This video builds the full definition from scratch: the magnitude formula, the right-hand rule for direction, the geometric interpretation as parallelogram area, and the algebraic properties that govern how cross products behave. Key concepts covered: • Magnitude formula: |A×B| = |A||B|sin(θ), with θ restricted to 0 ≤ θ ≤ π so the magnitude is never negative • Contrast with the dot product: the dot product uses cosine and returns a scalar; the cross product uses sine and returns a vector • Right-hand rule: redraw A and B tail-to-tail, curl the fingers of your right hand from A toward B, and your thumb points in the direction of A×B • The result is always perpendicular to the plane containing both vectors • Geometric interpretation: |A×B| equals the area of the parallelogram spanned by A and B, with either vector serving as the base • Parallel and anti-parallel vectors: when θ = 0 or θ = π, sin(θ) = 0, the parallelogram collapses, and the cross product is zero — including A×A = 0 • Anti-commutativity: A×B = −B×A; swapping the order reverses the direction, so order always matters • Scalar factoring: cA×B = c(A×B) and A×cB = c(A×B) • Distributivity over addition: (A+B)×C = A×C + B×C and A×(B+C) = A×B + A×C Next video: computing cross products component by component using Cartesian unit vectors. ━━━━━━━━━━━━━━━━━━━━━━━━ SOURCE MATERIALS The source materials for this video are from https://ocw.mit.edu/courses/8-01sc-cl...