Vector Cross Product & Triple Product Calculator

Vector Cross Product Guide

Cross Product Definition

For 3D vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃): a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). Determinant form: a × b = |i j k|/|a₁ a₂ a₃|/|b₁ b₂ b₃|. Direction: perpendicular to both a and b. Determined by right-hand rule: fingers from a to b, thumb gives direction of a × b. Magnitude: |a × b| = |a||b| sin θ. Where θ is the angle between vectors. The cross product is anti-commutative: a × b = −(b × a). a × a = 0 (vector with itself is zero). Parallel vectors: a × b = 0 (sin 0° = 0).

Geometric Interpretation

Magnitude of cross product = area of parallelogram spanned by a and b. ½|a × b| = area of triangle with sides a, b. Direction is normal to the plane containing a and b. Right-hand rule: point fingers along a, curl towards b, thumb points along a × b. Reverse: b × a points opposite. Applications in 3D: surface normals in computer graphics. Torque: τ = r × F. The angular momentum and torque are perpendicular to the radius and force vectors. Magnetic force on a current: F = IL × B. Force perpendicu

Triple Products

Scalar triple product: a · (b × c) = (a × b) · c. Geometrically: volume of parallelepiped with edges a, b, c. Equivalent to determinant: a · (b × c) = |a₁ a₂ a₃|/|b₁ b₂ b₃|/|c₁ c₂ c₃|. If zero: the three vectors are coplanar (parallelepiped has zero volume). Vector triple product: a × (b × c) = b(a · c) − c(a · b). Known as the BAC-CAB rule. Useful in deriving vector identities and solving differential equations. Note: cross product is not associative — (a × b) × c ≠ a × (b × c) in general.

Applications

Physics: torque τ = r × F (rotational equivalent of force). Angular momentum L = r × p. Lorentz force F = q(v × B) (force on moving charge in magnetic field). Engineering: moment of force about a point. Beam bending and structural analysis (calculate moments about supports). 3D Graphics: surface normal vectors for lighting calculations. Triangles in 3D: normal = (v₂ − v₁) × (v₃ − v₁). Backface culling: cross products determine whether a triangle faces the camera or away. Navigation: cross produc