The wedge product is better than the cross product | Geometric algebra episode 9

We are going to finish our discussion of three-dimensional geometric algebra today, and we're also finally going to get rid of the cross product. There's a much better alternative. But first, let's do a quick recap of two-dimensional geometric algebra so that you can see the similarities and the main differences. In a 2D vector space, by definition, we have two unit basis vectors. We can wedge them together into a little unit square, the basis bivector. We also have the real numbers, and that's all of the elements of our Clifford algebra. We get four basis elements in total, from which all other multivectors are constructed as linear combinations. We captured these numbers in the second row of the Pascal's triangle. Real numbers have grade zero. Vectors have grade one. And bivectors have grade two because they are constructed by wedging two vectors together. We discovered that the unit bivector squares to -1, and that it serves as the imaginary unit for a complex number system that consists of the even graded elements. In other words, each complex number has a grade zero real part and a grade two imaginary or bivector part. The unit bivector plays another important role, too. It's the pseudoscalar of the algebra, and we use it to find the Hodge dual for any element. For example, if you take one of the basis vectors, and you multiply it with the pseudoscalar I, it gets rotated 90°, which gives us the other basis vector. The two are indeed each other's duals, each other's orthogonal complement.

Great. So, now that we remember how everything works in 2D, we can easily extend everything to three dimensions. We just take the next row in the Pascal's triangle. We now have three unit basis vectors, and we can wedge them in pairs in three different ways to create the basis bivectors. Or, we can wedge all three of them together to create a unit trivector. Nothing new so far. The sum of the numbers in this row is eight. So, we get an eight-dimensional Clifford algebra. Multivectors are linear combinations of these eight basis elements. That's a lot of objects to play around with. And so, in 3D, we will get some new phenomena that weren't present in 2D. For example, we now have three unit bivectors that all square to -1. In the next video, we will use them to create a new number system, just like we created the complex numbers in 2D. The even graded elements will once again play the key role. Just like before, there is also a pseudoscalar. This time, it has grade three, so it's no longer a bivector. So, please keep in mind that we have two very different kinds of special objects now. The bivectors, which have grade two and square to -1 and serve as imaginary units, and the pseudoscalar W, which has grade three and helps us find Hodge duals. As I explained in the previous video, you get the Hodge dual of an object by just multiplying it with W. In 2D, each vector was Hodge dual to another vector. But this time, each vector is Hodge dual to a bivector. The two are each other's orthogonal complement. Each slice of a plane has its associated normal vector. This duality creates a lot of confusion, which we're going to spend the rest of the video unraveling.

We live in a three-dimensional world. We are so used to it that we don't even really think about it anymore. So, when we see a door opening, we would say that the door rotates around an axis, right? The axis of rotation runs along the hinges. Likewise, we would say that an ice skater makes a pirouette by rotating around a vertical axis. Or, the blades of a wind turbine rotate around a horizontal axis. This is so natural that we fail to realize that it's actually not the complete picture. Look at the complex numbers. They rotate inside their two-dimensional plane. There is no axis of rotation. There isn't enough room for such an axis. If you think about this a little longer, you will realize that rotations don't happen around an axis at all. They happen inside a plane. Rotations are fundamentally two-dimensional. The reason we are confused about this is because three happens to be a special number. You see, in 3D, when you rotate inside some plane, there's always exactly one direction that sits orthogonal to that plane. It's the Hodge dual. Vectors and bivectors are dual to each other. And so, it's easy to be confused and think of a rotation as happening around that normal vector, orthogonal to the plane in which the rotation actually takes place. In 4D, we have six basis bivectors, but only four basis vectors. Planes and vectors are no longer dual to each other. There is no unique axis anymore. Every plane of rotation now has two independent axes that are both orthogonal to it. To avoid all the confusion, it is better to think of a rotation as taking place inside a 2D plane, not around a 1D axis. This means that bivectors are the key to understanding rotations. We have already seen that the imaginary unit, the generator for all rotations, is indeed a bivector. So, from now on, whenever you see a rotating object or person, try to see the plane in which they rotate, rather than the axis around which they rotate. It takes a bit of practice, but you will get a new and interesting perspective on the world around you.

We have one more confusing piece of vector algebra to deal with. It's called the cross product. You may have heard of it, or you may have used it in vector calculus or in physics. The idea is that you have two vectors, U and V, and their cross product is a third vector, W. This new vector is perpendicular to both U and V, and its length is defined to be the area of the parallelogram between U and V. You can calculate it using this weird determinant of a matrix that contains U and V as rows, but also contains a row filled with the three basis vectors. If you spell out the entire determinant, you get this formula. And basically, this is everything you are typically taught about the cross product. This definition is weird. It feels artificial. Where did the determinant come from? Why is the result orthogonal to the two input vectors? Why is the length of the resulting vector equal to an area? That doesn't even seem to type check. What's going on here? In addition, the cross product has some crazy and annoying properties. It's not commutative, but it's actually not even associative. And it exists only in a three-dimensional vector space, because that's the only space in which the determinant has the correct number of rows and columns. Now, don't get me wrong. The cross product is definitely useful.

When you apply a force in a rotating arc, such as when you pull open a door, the rotational force is called the torque. It's typically defined as the cross product between the force vector and the vector that runs from the hinges to the door handle. But something feels wrong about this. I mean, the door opens by rotating in the plane containing the two vectors. There is no physical reason for the resulting torque to be orthogonal to that plane. The orthogonal direction doesn't participate in the rotation at all. But it gets even crazier. When you look at the situation in a mirror, the reflected door opens in the opposite direction. This time, the torque vector points down, which is not what an ordinary vector would do if you reflected it in a mirror. This strange behavior is why physicists call the output of the cross product an axial vector, named after the axis of rotation that runs through the hinges. They also often call it a pseudovector. And that is a name that should ring a bell. We will fix all the issues with the cross product in a moment. I first want to give you a second example, just to show you how ubiquitous the cross product and its weird behavior are in physics. When you run an electric current through a wire in the shape of a loop, this produces a magnetic field that runs through the loop. The field is often depicted as one or more vectors pointing orthogonally to the plane of the loop. You calculate the magnetic field with a cross product. And again, if you reflect this physical setup in a mirror, or if you just change the direction of the current in the wire, the magnetic vector flips upside down. It's an axial vector or a pseudo vector, just like the torque.

You have probably already guessed how we can fix all this craziness. The solution is basically already staring at us in these diagrams. Since the torque and the magnetic field are both related to rotations, we should find the plane in which those rotations take place. In the opening door example, the two vectors both lie in the rotation plane. So, if we just take their wedge product, we get a bivector that perfectly represents that plane. It captures all of the important information. No need for an orthogonal axis, and no more weird flipping upside down. In the mirror, the bivector simply has the opposite orientation. And that is precisely what you would expect. It's much more natural. The same is true for the magnetic field. It's not a weird kind of pseudo vector, it's a bivector. After reflection, it just gets the opposite orientation, and it looks totally natural in the mirror. So, what we really need is not the cross product, but the wedge product. We don't want a pseudo vector, we want a bivector. It just so happens that in 3D, and only in 3D, both of these views are dual to each other, because each plane has a unique vector or axis orthogonal to it, and vice versa. And that is where all the confusion and weirdness comes from.

We can even prove algebraically that the cross product and the wedge product are Hodge duals of each other. We just use the same old strategy that we always use. Write out the two vectors, and distribute the sums over each other. Apply the rules for the wedge product of basis vectors, and you get this formula. It already looks suspiciously similar to the formula we had for the cross product. The coefficients are already almost the same. The biggest difference is that the cross product uses vectors, while the wedge product uses bivectors. And we already know how to fix that. All we have to do is multiply with the pseudo scalar double you. Just look at how it perfectly gets rid of all the bivectors, and turns them into the correct vectors. Other than the leading minus sign, the two formulas are now exactly the same. This is brilliant. It means that we can throw the cross product in the garbage, because we can always use its dual instead, the wedge product. And why, you might ask, is the wedge product so much better? Let me count the ways.

The cross product produces vectors that flip around in a mirror. The wedge product produces bivectors that are perfectly well-behaved. The cross product exists only in 3D. The wedge product is much more general, and it can be defined in any number of dimensions. Unlike the cross product, the wedge product is associative. That makes it much easier to calculate with. It means that you can wedge three or more vectors together, because the entire point of associativity is to turn binary operations into n-ary ones for any value of n. We have an entire video about that. Also, the wedge product can be defined between any types of objects, not just vectors. You can wedge a bivector to a trivector to obtain a five vector, for example. You just add up the grades. The cross product is only defined for vectors. But to me, the most important advantage is that the cross product was giving us the wrong picture. It tricked us into believing that we rotate around an axis. The truth is that rotations take place in planes. They are essentially two-dimensional. This is a more consistent picture, which once again generalizes easily to higher dimensions, or to lower ones where there is no room for an axis. So, if there is only a single lesson from geometric algebra that should make it into all textbooks, I vote for this one. Please forget the cross product, and use the wedge product instead from now on. Also, please like and subscribe, and support us on Patreon. I will see you next time when we will discover the quaternions hiding inside 3D geometric algebra.