crossproducts

Cross products, or the vector cross product, is a binary operation in three-dimensional space that takes two vectors and returns a third vector perpendicular to both inputs. For vectors a = (a1, a2, a3) and b = (b1, b2, b3), the cross product is a × b = (a2 b3 − a3 b2, a3 b1 − a1 b3, a1 b2 − a2 b1). The resulting vector is orthogonal to the plane containing a and b, with direction given by the right-hand rule. Its magnitude equals |a||b|sin θ, where θ is the angle between a and b, so the cross product is zero when the vectors are parallel or one is the zero vector.

Key properties include bilinearity and antisymmetry: a × b = −(b × a). The cross product is only defined in three-dimensional space (in the standard sense); higher-dimensional analogues exist in the context of exterior algebra and via special constructions, but no direct vector-valued cross product with all the same properties exists in general n dimensions.

Computational note: the cross product can be computed using the determinant of a 3×3 matrix with i, j, k as unit vectors, yielding the components shown above. Geometrically, |a × b| equals the area of the parallelogram spanned by a and b.

Applications span physics, engineering, and computer graphics. In physics, cross products model torque τ = r × F and angular momentum L = r × p. In computer graphics, they are used to compute surface normals for lighting calculations. In geometry and vector calculus, they provide a convenient measure of orthogonality and orientation of vectors.