Cross Product
Definition
A way to multiply vectors that gives a new vector perpendicular to both — the math behind torque and magnetic force.
Symbol: · Unit: depends on inputs
Technical definition
For vectors a and b, a × b has magnitude |a||b|sinθ and direction given by the right-hand rule. Anticommutative: a × b = −(b × a). Appears in torque (τ = r × F), angular momentum (L = r × p), and magnetic force (F = qv × B).
Example
Torque τ = r × F: pushing a door at right angles (θ = 90°) maximizes torque; pushing straight at the hinge (θ = 0°) gives zero — the sine captures the geometry.
Common confusion
Cross product is not commutative — a × b points opposite to b × a. Order matters, and the result is a vector (unlike the dot product's scalar). Mixing them up breaks torque and magnetism problems.