Dear Cross Product,
I know I’ve said some pretty nasty things about you in the past. I won’t rehash the entire list here, but I will try to explain the feelings that brought about my unjustified words.
My frustration began when I first learned the name given to you by Josiah Willard Gibbs. He named you a “product” and that naturally made me think of the concept of multiplicative product. After considerable thought, I’ve concluded that you’re not actually a product at all. You’re a geometrical construct that has nothing to do with multiplication. For years, I’ve been glossing over this detail with my students, not that they caught onto the problem. If they did, they never let on. Nevertheless, it bothers the heck out of me now. I’ve been accused of being needlessly pedantic, but I’d rather be accurate than sloppy when terminology is involved. Just think how we (physics teachers) screw with students’ minds with “massless rope” or “frictionless pulley” or “infinitely far away.” These are all well intended terms, but none is physically accurate. I think I’m going to call you by a new pet name, like maybe just “vector cross.” Honestly, it’s a term of endearment, and one that removes all hints of multiplication. Yeah…it fits. Like it? Oh by the way, I’m going to start calling your sibling “vector dot” instead of “vector dot product” too.
My frustration continued when I realized that you exhibit a geometrical paradox, or so I thought. The two vectors out of which you’re constructed define a plane, but somehow out of that combination we get something that is perpendicular to that plane! Ridiculous! When a see-saw rotates one way or the other under the influence of a net torque, the rotation happens in the plane defined by the moment arm and the applied force. The physics happens in a plane, not in a direction perpendicular to that plane. If that were really true, then there would be no legitimate way to talk about torque or angular momentum in two dimensional problems. This really bugged me until I finally figured out where you’re coming from. It dawned on me that your direction is defined to coincide with the axis around which rotation happens. I get that now, but it would help if this were explicitly documented somewhere, especially in the introductory literature. This whole “how you get something from a plane that’s perpendicular to that plane” thing really got to me and made me say you should “die a quick death.” I’m really sorry I said that, but it’s what I felt at the time.
WAIT! DON’T GO! HEAR ME OUT….PLEASE!
Anyway, all that’s in the past now. I’ve grown. I’ve matured. I now realize that you are wise beyond your years. I’ve changed, I want to tell you why. See, I have this weird fantasy that introductory calculus-based physics should include elementary (VERY elementary) aspects of tensor analysis, at least enough to give students an understanding of the inertia tensor in certain simple geometries. Well, you’re the secret ingredient to making that happen. Your ability to be written as both a vector and an anti-symmetric tensor, along with a review of elementary linear algebra students have already seen in previous classes, is the bridge from vectors to tensors. You ARE a tensor, in a weird sort of way.
But there’s something else about you that I never appreciated before. You’re necessary to discuss a vector’s derivative. You’re the best way to describe the change in a vector when the vector’s magnitude is constant. I was aware of this years ago, but the underlying logic escaped me until very recently. Now I see it all.
I guess that’s all I wanted to say. I hope you can forgive me.