# Quick insights from twitter into vector products

I love twitter specifically because it presents me with so many ideas, even new ways about thinking about old things or ideas I thought I already understood. Here’s today’s example, courtesy of Superfly Andy Rundquist:

and Joe Heafner promptly responded with

I don’t know about you, but my first introduction to dot and cross products was filled with trying to understand i,j,k notation and follow weird procedures for manipulating vectors in a matrix form, and I had no clue what I was doing, nor did I understand the significance of what I was calculating.

How great would it be to simply introduce the cross product to a class with Andy’s question? Stand next to a kid, ask “how much of you is perpendicular to me?” Then lean over to the side at a 10 degree angle and ask the question again. Students could probably measure this with a meter stick. Later try all sorts of other situations, like clock hands, and later, more abstract ideas like position and momentum vectors.

This would have helped 19 year old me out tremendously.

Awesome, I’m glad Joe and I could provide some inspiration. Recently I’ve been teaching about plane wave with things like and we’ve been trying to figure out why they’re called plane waves. If you draw an arbitrary r vector and ask “hey you, how much of you is parallel to k?” then the next question is “hey, what’s another r that would give the same answer”. Boom, the locus of all r’s that answer that form a plane. Fun!

It’s amazing how much physics follows from these two bits of geometry! I can’t resist sharing another slightly more sophisticated gem with you! We are usually conditioned to think of dot products and cross products as separate and distinct entities, but they can actually be seen together in a little used “thingy” called a dyadic product. It’s rather like a Gibbsian analogy to the geometric product from geometric algebra in that it encodes “parallelness” and “perpendicularness” (yes, I made those terms up) into one entity. Think of a vector as a trinomial, and then multiply two such vectors together term by term without treating the cross terms as commutative, and you get terms that can be arranged in a 3×3 matrix. For example, multiplying (ax e_x + ay e_y + az e_z) by (bx e_x + by e_y + bz e_z) where the e’s are basis vectors (aka ihat, jhat, khat) gives terms like ax bx e_x e_x + ax by e_x e_y + ax bz e_x e_z + … + az bz e_z e_z. If you arrange these terms into a 3×3 matrix, the diagonal contains the terms of a dot b and the off diagonal terms are those of a cross b (you have to cross the diagonal to get one term)! You can explicitly see the antisymmetry of a cross b by visual inspection! I’ve yet to see this exploited in the introductory literature. You’ve also just crossed into tensors! Geometry! Unification!

How does asking the question “How much of you is perpendicular to me?” help reach conceptual understanding of the direction of the cross product?

It doesn’t. For that you need the dreaded right hand rule, of course.

Those two vectors define a plane, and the order of the vectors defines an orientation. The right hand rule is just a choice of orientation.

Maybe it is just my dirty mind, but I wouldn’t use “hey you, how much of you is perpendicular to me?” in a high school classroom.