Cross product calculator
Calculate the cross product a × b of two 3D vectors. Shows the determinant method step by step.
Vector a
Vector b
Cross product formula
a × b = |i j k; a₁ a₂ a₃; b₁ b₂ b₃|
= (a₂b₃ − a₃b₂)i − (a₁b₃ − a₃b₁)j + (a₁b₂ − a₂b₁)k
Properties
The cross product is a vector perpendicular to both input vectors. Its magnitude equals the area of the parallelogram formed by the two vectors. It's only defined in 3D.
Anti-commutative: a × b = −(b × a)
Magnitude: |a × b| = |a||b|sin(θ), where θ is the angle between them.
Applications
Physics: Torque τ = r × F. Magnetic force F = qv × B.
Geometry: Normal vector to a plane, area of a triangle in 3D.
Graphics: Surface normals for lighting calculations.
Popular slope & distance problems
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