The wedge product can be thought of as what the vector product of vector analysis should have been. If are the unit vectors along the three coordinate axes in 3-dimensional euclidean space and
are two vectors, then the calculus student is told that
Unfortunately for people who want to do more complicated things (for example, relativity theorists working in 4-dimensional space-time), the vector product only works — is indeed only defined — in three dimensions.
But the wedge product of vectors, , works in any dimension, and the vector product, , can be thought of, in a certain sense, as a special case or modification of . The trick is that if you form the wedge product of vectors in the -dimensional euclidean space , the result is not a vector in but rather a “vector” in a different vector space .
To gain some feeling as to what this new “vector” is, recall that a vector in the plane or in 3-dimensional space is often thought of as a directed line segment. If and are vectors in , then they determine a parallelogram having and as its sides. We then identify with this parallelogram. More precisely, we identify it with an equivalence class of such parallelograms, and we also equip these parallelograms with a sort of “right-handedness” or “left-handedness,” what a mathematician likes to call an orientation.
The process can be extended. We can form “simple 3-vectors” which we can think of as “oriented” 3-dimensional parallelepipeds and then go on to even higher-dimensional “simple -vectors” . We can multiply them by scalars and add them and use them to do analytic geometry and calculus on curved surfaces in -dimensional euclidean space.
The reader who wants to know more is invited to click on The Wedge Product and Analytic Geometry and download a pdf file of the article which appeared in the August-September, 2008, issue of The American Mathematical Monthly. This article, written by Mehrdad Khosravi and yours-truly, presumably has at least a modicum of value since it won a Lester R. Ford Award for exposition.