II.8 - CROSS PRODUCT IN FOUR DIMENSIONS AND BEYOND

Abstract

Publisher Summary This chapter discusses cross product in four dimensions and beyond. Cross product is one of God's great gifts to mankind. It has many applications in mathematics, physics, engineering, and, computer graphics. Normal vectors, rotations, curl, angular momentum, torque, and magnetic fields make use of the cross product. Given two linearly independent vectors u and ν in three dimensions, their cross product is the vector u × ν, perpendicular to the plane of u and v and oriented according to the right-hand rule. There is a way to look at the cross product that is more instructive than the standard definition and that generalizes readily to four dimensions and beyond. The tensor product is closely related to the dot product. Like dot product, the tensor product makes sense for two vectors of arbitrary dimension. Indeed, the tensor product shares many of the algebraic properties of the dot product. The wedge product of two vectors u and ν measures the noncommutativity of their tensor product. Like the tensor product, the wedge product is defined for two vectors of arbitrary dimension. The wedge product also shares some other important properties with the cross product.