The Wedge product is the multiplication operation in exterior algebra. The wedge product is always antisymmetric, associative, and anti-commutative. The result of the wedge product is known as a bivector; in (that is, three dimensions) it is a 2-form. For two vectors u and v in ℝ3, the wedge product is defined as
where ⊗ denotes the outer product. Note that the bivector has only three indepedent elements; as such, it can be associated with another vector in ℝ3. If the associated vector is defined as
it is the same as the cross product of u and v. In this sense, the cross product is a special case of the exterior product.
- See also: Rotation_matrix#Rotation_matrix_from_axis_and_angle and Cross_product#Conversion_to_matrix_multiplication
Bivectors are commonly used to represent rotations.