Every undergraduate in science or engineering has learned to add the concept of vectors to his repertoir, expanding mathematics beyond the application of numbers only. Vectors are liberating; they form a tremendously powerful approach to physics, constituting the backbone of almost all science and engineering today. A vector is defined as a little arrow, i.e., a directed line segment. The addition of vectors is defined in a natural and physically intuitive way by moving one vector to the top of the other. Then draw the vector that starts at the bottom of the first and ends at the top to the second. That's the sum of the original two.
Multiplication of vectors, by contrast, is problematic. There are two ways of "multiplying" vectors, neither of them philosophically satisfactory. The dot product, v ⋅ w, and the cross product, v × w, are both problematic. The dot product takes you out of the realm of vectors into the realm of scalars and the cross product only exists in dimension 3. Worse, it takes what is an essentially 2-dimensional situation, a relationship between two vectors which are sitting in the same plane, and pops you up to an apparently extraneous third dimension.
The multiplication of vectors is so unsatisfactory, in fact, that in most treatments of vectors in higher mathematics it is ignored entirely, vectors being thought of merely as objects which it is possible to add (and stretch), but not as a general rule to multiply.
There is a new way to look at multiplication between vectors however, a new foundation of vector relations which allows the laws of physics to be rewritten much more succinctly and intuitively. The name is "Geometric Algebra", and detailed references can be found here, here, here, and here. The new formalism appears to be very powerful, allowing such disparate topics as Maxwell's equations, the Pauli spin matrices familiar from quantum mechanics, and relativity theory, for three examples, to be reformulated much more clearly than ever before. Maxwell's equations can now be reduced to a single line as shown below, for example.
∇ F = JThat single line contains all of Maxwell's equations in one fell swoop!
Like all revolutionary acts of brilliance, the fundamental ideas of geometric algebra are quite simple yet very powerful.
The first idea is to allow a few more objects than simply scalars and vectors into our theory. We now also include directed areas (parallelograms), called bivectors. "Directed" in this case means that we have chosen either a clockwise or a counterclockwise direction to traverse the parallelogram, and associated it to the area. If we start with the vectors v and w, then we define the bivector
v ∧ wto be the parallelogram formed by moving w along v, with the orientation taken to be that given by first moving along v and then along w. This new product, called the wedge product, is an area normal to the cross product, having area equal to the length of the cross product. It is defined in every dimension and doesn't remove us from the 2-dimensional situation we started with. It is in fact the replacement for the cross product in the new formalism. The wedge product is antisymmetric, i.e.,
v ∧ w = - w ∧ v,(because the bivector on the left is oriented clockwise while the one on the right is oriented counterclockwise), whereas the dot product is symmetric,
v ⋅ w = w ⋅ v.This first new idea, bivectors, can be likewise generalized to trivectors, a trivector being an oriented parallelepiped, and similar oriented objects in higher dimensions.
The second great insight of the new scheme is that the multiplication of vectors should contain both a symmetric and an antisymmetric component at the same time. Therefore a new product, the geometric product of two vectors is defined by the simple formula
vw = v ⋅ w + v ∧ w.This definition still suffers from the drawback that a pair of 1-dimesional objects (vectors) has given rise through "multiplication" to a 0-dimensional object (a scalar) and a 2-dimensional object (a bivector). In a sense, this seems to balance out and to be more satisfying than either the dot product or the cross product alone. But the real solution is to change the scope of operations. Instead of considering vectors as our fundamental objects, we now consider multivectors to be the fundamental objects of mathematical physics, where multivectors are simply abstract sums of the form
a + bu + c(v ∧ w),with a, b, and c being ordinary scalars, u an ordinary vector, and v ∧ w a bivector. As with the definition of the complex numbers, the disparate pieces of the multivectors are added only with like pieces, allowing us to stay in the arena of multivectors when adding, and the geometric product defined above allows us to remain in the same arena when multiplying.
In dimension 3, the proper objects are now considered to be the set of all multivectors of the form
a + bu + c(v ∧ w) + d(s ∧ t ∧ r),with a, b, c, d real numbers. Similarly, in dimension 4 we throw in a quadvector term on the end.
Experience has painfully taught us that the best long-term progress in our theories comes from improvements in our fundamental notations themselves, in what mathematicians call the "machinery" which we use to conceptualize reality. Geometric Algebra seems to hold the potential to be an earth-shattering paradigm shift in physics, engineering, and mathematics. Keep an eye on it.
Update: Graphics for wedge and dot products fixed as per Seneca's suggestion, using the code found on this useful page.