(Cover Artist: Sophie Cheng)
The Dot and Cross Products:
Some of you may have heard of vectors before. If you have, your first exposure has probably been in a precalculus or physics class, where vectors are defined as “arrows” or oriented lengths: objects with both magnitude and direction. This geometrical portrait of vectors is certainly very useful, and we will be utilizing it in order to gain an intuition for geometric algebra. If you’ve learned about vectors, you may also know about the two main vector operations: the dot product and the cross product. I’ll start with the dot product, the simpler one of the two, to begin. In short, the dot product measures how much two vectors point in the same direction. You can see this property in one form of the dot product:
Note the cosine factor in this expression; cosine is greatest when the angle between the vectors is equal to 0 degrees (both vectors are pointing in the same direction), equal to zero when the angle between the vectors is equal to 90 degrees (the vectors are perpendicular), and least when the angle between the vectors is equal to 180 degrees (the vectors are pointing in opposite directions).
While the dot product is pretty simple, the cross product is much more annoying to deal with. Roughly speaking, the cross product measures how perpendicular the two vectors are instead of how parallel they are. Like with the dot product, this property is evident from an expression for the cross product, or rather, the magnitude of the cross product:
In contrast to cosine, sine is maximized when the two vectors are perpendicular and zero when the two vectors are parallel. Those of you who have a fresh memory of geometry class may also notice that uvsinθ is the formula for the area of a parallelogram, specifically the area of the parallelogram formed by the two vectors, which will be important later.
The important difference between the dot and cross products is that the cross product is a vector quantity. (Actually, the cross product of two vectors isn’t technically a vector, it’s an object called a pseudovector, but that is beyond the scope of this article. )
If we think of the two vectors as being on a plane, then the cross product of the two vectors is in the direction perpendicular to that plane (this direction can be defined with a vector: the unit normal vector (perpendicular to a surface), with a magnitude given by the formula I introduced earlier). Keen readers might notice that this property leads to an issue: there are two possible directions that this vector could point in. The tiebreaker here is the right hand rule. If you point your right hand pointer finger in the direction of the first vector in the product, your middle finger in the direction of the second vector, your thumb should point in the direction of the resulting vector. Note that this decision is completely arbitrary; we could just as easily have defined a left hand rule, which flips the orientation of the cross product compared to the right hand rule (try it for yourself!).
All of these properties of the cross product make it fairly unwieldy to use. There’s not a particularly easy way to find the exact direction of the cross product vector. Finding the magnitude is also easier than it sounds, at least compared to the dot product, since to find the angle, you will likely have to compute a dot product anyways. The other formula for the cross product requires you to compute a 3D determinant, which is pretty annoying in and of itself. The cross product is also dimension limited. The 3D determinant method is restricted to (surprise!) three dimensions. The problem, however, is even more fundamental than that. The purpose of the cross product is to find a vector perpendicular to two other vectors. Not only does this task not make any sense at all for lower dimensions, but there are infinitely many possible vectors that satisfy this condition for higher dimensions.
For lower (< 3) dimensions, you can justify this statement to yourself by simply noticing that there is no “space” for a third vector to be perpendicular to two vectors at the same time. For obvious reasons, there isn’t really a good intuition for higher (> 3) dimensions. With a basic understanding of linear algebra, you can justify this statement to yourself a little bit more rigorously (Hint: what vector spaces can three vectors be the basis vectors of? What vector spaces are spanned by three vectors? In which vector spaces are three vectors linearly independent?).
A simple right hand rule will not fix that. So, you might ask, is there a better way to accomplish the task of the cross product?
Yes, there is!
Our replacement for the cross product is the exterior product. (Not to be confused with the outer product — it’s zany, I know.) Instead of introducing a perpendicular vector as a middle-man, the exterior product uses the two vectors
themselves. Remember the parallelogram from earlier? The exterior product is that parallelogram, or rather, it almost is. We still need to deal with the problem of orientation. The orientation of the exterior product of two vectors is defined as beginning in the same direction as the first vector and “curling” in the direction of the second vector. Exterior products give way to two orientations: clockwise and counterclockwise. It’s a bit easier if you first see an example. In the diagram below, the orientation of the exterior product is clockwise.
Note that if we flipped the order of the exterior product, i.e. v ∧ u, the orientation would be counterclockwise. This fact clearly demonstrates the anti-commutativity of the exterior product (this property states that u ∧ v = -v ∧ u). This property is called anti-commutativity because rather than the exterior product being commutative, it ends up being the negative. The cross product also has this property, but it is much less obvious, another reason why the exterior product is superior. The magnitude of the exterior product is much the same as the magnitude of the cross product, it’s equal to its area. By the way, the exterior product of two vectors is called a bivector. A bivector, conveniently, can be described as an oriented area just like how a vector can be described as an oriented length.
There are a couple of other properties of the exterior product and the bivector that make it very convenient to use. One of these properties is the existence of basis bivectors. In the Cartesian plane, we can define one set of basis bivectors as all possible exterior products of basis vectors, which would be i ∧ j, j ∧ k, and i ∧ k. All other possible combinations either equal 0, such as i ∧ i, or can be easily expressed as other basis bivectors, such as k ∧ j. Very conveniently, all simple bivectors, i.e. bivectors that can be expressed as the exterior product of two vectors, can be expressed as a linear combination of basis bivectors. Just like there are basis vectors that come in handy when doing vector algebra, basis bivectors can help us with doing algebra with bivectors. Case in point, take the following example of the exterior product of two vectors:
Note that we used two properties of the exterior product that I haven’t mentioned yet: distributivity over addition and the multiplication of magnitudes. However, it should be pretty easy to convince yourself of these properties if you just go back to picturing the exterior product as a parallelogram. The exterior product is also associative, while the cross product is not — it instead satisfies the Jacobi identity. Try to justify this property to yourself (Note: you may need to use the exterior product of three vectors, which is called a trivector and can be thought of as an oriented volume). All of these characteristics of the exterior product make it easily more convenient to use than the clunky cross product.
Geometric Product (for Vectors) As a conclusion for our tour into the world of exterior algebra, we look at the geometric product, which is a type of product defined on vectors, like the dot, cross, and exterior products. The geometric product can also be defined for bivectors, trivectors, etc. and combinations of vectors, bivectors, trivectors, etc. (collectively called multivectors). However, this generalization is beyond the scope of this article, but you can consult a geometric algebra textbook if you would like to learn more.
However, it is built out of components we are already familiar with, namely the dot and exterior products:
This operation is particularly useful because it has two distinct components: a scalar component given by the dot product and a bivector component given by the exterior product. This means that with one product, we can encode two pieces of information about the two vectors that went in. One application of this property is the ultra-condensed form of Maxwell’s equations:
You might be wondering: how can we express four equations, moreover, four fairly complicated equations, as just one equation? Well, with the power of geometric algebra! ∇F = J is something called a multivector, which is a combination of scalars, vectors, bivectors, trivectors, etc. Think of a multivector like a complex number, which is just the sum of a real number and an imaginary number. Well, each of Maxwell’s equations corresponds to a specific type of quantity. So, by simply taking the scalar, vector, bivector, or trivector component of this unified equation, each of the four Maxwell equations that went into it can be recovered. Isn’t that amazing?
Of course, there is still a lot to be explored that I couldn’t quite get to in this article, such as applications to calculus, rotations, and complex numbers. However, I hope that I’ve given you a nice, snappy introduction to exterior products and geometric algebra that was, at the very least, an interesting excursion.