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Cross product
Intermediate
Vector product perpendicular to both, magnitude equals parallelogram area.
∣A×B∣ = AB sinθThe cross (vector) product A×B has magnitude AB sinθ and a direction perpendicular to both, given by the right-hand rule. It is largest for perpendicular vectors and zero for parallel ones. Torque τ = r×F and magnetic force qv×B are cross products.
î × ĵ = k̂. A force of 10 N perpendicular to a 0.5 m wrench gives torque 5 N·m along the axis of the bolt.
The order matters: A×B = −(B×A).
In more detail
The cross product of two vectors gives a third vector perpendicular to both, with magnitude AB sin θ and direction given by the right-hand rule. It is used for torque, angular momentum and magnetic force.
Real-world uses
- Torque is the cross product of the lever arm and the force, so a wrench turns most when pushed at right angles.
- The magnetic force on a moving charge is the cross product of velocity and magnetic field.
- The torque on a wrench is the cross product of the lever arm and the force, so only the perpendicular part of the force turns the bolt.
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Frequently asked questions
What is the direction of a cross product?
Use the right-hand rule: point the fingers along the first vector and curl them towards the second, and the thumb gives the direction.
Is the cross product commutative?
No. Reversing the order reverses the direction, so A × B equals minus B × A.
How do I remember which way a cross product points?
Use the right-hand rule: curl your fingers from the first vector toward the second, and your thumb points in the direction of the result.
Related topics in Measurement & Math Tools
Vector components · Unit vectors · Dot product · Triple products · Calculus: derivatives · Calculus: integrals