X×V

X×V denotes the cross product of two vectors X and V, most commonly used in three-dimensional space to produce a vector perpendicular to both X and V. In standard coordinates, if X = (x1, x2, x3) and V = (v1, v2, v3), then X×V = (x2 v3 − x3 v2, x3 v1 − x1 v3, x1 v2 − x2 v1).

The magnitude of the cross product satisfies |X×V| = |X||V| sin θ, where θ is the angle between X and V. The direction of X×V is determined by the right-hand rule: curling the fingers from X toward V, the thumb points in the direction of X×V. The vector X×V is orthogonal to both X and V.

Key properties include anti-commutativity (X×V = −(V×X)) and distributivity over addition (X×(Y+Z) = X×Y + X×Z). The operation is bilinear and linear in each argument. The cross product is particularly important in physics and engineering for torque (τ = r × F) and angular momentum (L = r × p), and in electromagnetism for the magnetic force on a moving charge (F = q v × B).

Limitations and variants: the standard cross product is defined for two vectors in three-dimensional Euclidean space (with the usual orientation). It is not a universal binary operation in all dimensions; higher-dimensional analogs exist but behave differently. In computational contexts, X×V is often implemented with explicit coordinate formulas as above.