×v×B

The expression v × B denotes the cross product of velocity v and magnetic field B in three-dimensional space. The resulting vector is perpendicular to both v and B, with magnitude |v||B|sinθ, where θ is the angle between them. The direction is given by the right-hand rule: if the fingers sweep from v toward B, the thumb points in the direction of v × B.

In physics, the cross product v × B appears in the magnetic part of the Lorentz force law, F = q(E + v × B). Here q is the electric charge and E is the electric field. The magnetic force q(v × B) is always perpendicular to the particle’s velocity, so it does no work on the particle and changes only the direction of motion, not its speed. In a uniform magnetic field, a particle with velocity perpendicular to B moves in a circle with cyclotron frequency ωc = qB/m and radius r = mv/(qB); if there is a velocity component parallel to B, the motion is helical.

Mathematically, in Cartesian coordinates, v × B = (v_y B_z − v_z B_y, v_z B_x − v_x B_z, v_x B_y − v_y B_x). The cross product is bilinear and antisymmetric: a × b = −(b × a) and a × (b + c) = a × b + a × c. It is a fundamental operation in electromagnetism, with applications ranging from particle dynamics in magnetic fields to the design of electric motors and magnetic confinement systems.