Anticrossproducts and cross divisions

Abstract

This paper defines, in the context of conventional vector algebra, the concept of anticrossproduct and a family of simple operations called cross or vector divisions. It is impossible to solve for a or b the equation a×b=c, where a and b are three-dimensional space vectors, and a×b is their cross product. However, the problem becomes solvable if some “knowledge about the unknown” (a or b) is available, consisting of one of its components, or the angle it forms with the other operand of the cross product. Independently of the selected reference frame orientation, the known component of a may be parallel to b, or vice versa. The cross divisions provide a compact and insightful symbolic representation of a family of algorithms specifically designed to solve problems of such kind. A generalized algorithm was also defined, incorporating the rules for selecting the appropriate kind of cross division, based on the type of input data. Four examples of practical application were provided, including the computation of the point of application of a force and the angular velocity of a rigid body. The definition and geometrical interpretation of the cross divisions stemmed from the concept of anticrossproduct. The “anticrossproducts of a×b” were defined as the infinitely many vectors xi such that xi×b=a×b.