A×B

A×B is a notation used in mathematics to denote more than one binary operation, depending on the context. The most common meanings are the Cartesian product of two sets and the cross product of two vectors in three-dimensional space.

In set theory and related areas, A×B denotes the Cartesian product of sets A and B. It is the set of all ordered pairs (a,b) where a is an element of A and b is an element of B. The size of A×B is the product of the sizes of A and B, i.e., |A×B| = |A||B|. If either A or B is empty, A×B is empty. This construction underpins many definitions in mathematics, such as relations, functions as special subsets of A×B, and product spaces in topology. For example, if A = {1,2} and B = {x,y}, then A×B = {(1,x), (1,y), (2,x), (2,y)}.

In vector mathematics, A×B denotes the cross product of two vectors A and B in three-dimensional space. The result is a vector perpendicular to both A and B, with magnitude |A||B|sinθ, where θ is the angle between A and B. The components are given by A×B = (AyBz − AzBy, AzBx − AxBz, AxBy − AyBx). The cross product is bilinear and anti-commutative (A×B = −B×A); it follows the right-hand rule for direction. It is defined specifically for 3D vectors (with generalizations in other contexts).

Notes: some authors use A×B to indicate a matrix-related product in particular contexts, but the standard matrix product is usually written AB. The exact meaning of A×B depends on the mathematical setting.