orthogonal to it through the left contraction with a suitable bivector. In two dimensions this is a rotation through a right angle. As a further generalization, we can loosen the requirements of multilinearity and magnitude, and consider a general continuous function (where is endowed with the Euclidean inner product and ) which is only required to satisfy the following two properties: The cross product is always orthogonal to all the input vectors. If the input vectors are linearly independent, then the cross product is nonzero. Under these requirements, the cross product only exists (I) for , (II) for , (III) for , and (IV) for any . See also Composition algebra Notes References Also available as ArXiv reprint . Bilinear maps Octonions Operations on vectors