The cross product of two vectors is notjust another vector ‐‐ a major misconception being perpetuated in calculus and vector analysis textbooks

Abstract

The vast majority of calculus textbooks perpetuate the misconception that the cross product of two vectors is simply another vector. This is despite the fact that scientists have known for over a century that such a representation creates major inadequacies in our description of various branches of physics. Unless it is our intent to force students to unlearn false ideas when they progress to higher levels of mathematics ‐‐ namely, the cross product of vectors does not transform according to the laws of tensors ‐‐ a much different picture should be presented. Fortunately, instead of the traditional short‐sighted definition, the cross product of two vectors can be easily explained, by starting from the perspective of dyadics. The concept of multiplication of vectors now has a simple geometrical picture that encompasses both the dot and cross products in any number of dimensions in terms of orthogonal unit vector components. By this approach the limitation of such an entity to exactly a three‐dimensional space that does not allow for one of the three metric motions (reflection in a mirror) may be discarded and an appreciation of the intrinsic difference between true vectors and pseudo‐vectors is developed.