rýp

r×p denotes the cross product of two vectors r and p in three-dimensional space. The result is a vector perpendicular to both r and p, with magnitude equal to the area of the parallelogram spanned by r and p. It can be computed as r×p = (r2 p3 − r3 p2, r3 p1 − r1 p3, r1 p2 − r2 p1). The direction is given by the right-hand rule: if the fingers of the right hand curl from r toward p, the extended thumb points in the direction of r×p. Its magnitude is |r×p| = |r||p| sin θ, where θ is the angle between r and p. The cross product is bilinear and antisymmetric: r×p = −p×r and r×(p+q) = r×p + r×q.

In three dimensions, the cross product can be written using the Levi-Civita symbol or as a determinant with the standard basis vectors i, j, and k. It is not generally defined in the same way in higher dimensions, though there are related generalizations such as the exterior (wedge) product. The cross product yields a pseudovector (axial vector) that transforms differently under improper rotations than ordinary vectors.

Common applications include torque τ = r×F, angular momentum L = r×p, and the magnetic force F = q v×B in electrodynamics. In computer graphics and physics, the cross product is used to determine orientations, rotational axes, and perpendicular directions in 3D space.

In two dimensions, a true 3D cross product is not defined for planar vectors, but one may embed 2D vectors in 3D by adding a zero z-component to compute a z-axis cross product component, whose magnitude equals the parallelogram area.