23rd
Vectors
Annie, Geddes, and I have decided to take an introductory physics class on MIT opencourseware this winter, and we’re still in the kind of background info about math and measurement and stuff. Which means I spent an hour talking to Geddes about vectors today. What fun!
I’d like, first of all, to offer you this opportunity to look up “dot product” and “cross product” on wikipedia and spend a few moments pondering those delightful operations. That’s basically what we did, and it was pretty nice. High points: the dot product gets bigger the closer the vectors are to parallel; there is something parallel-y about cosine-y-ness! And (parallelly, pun intended) there is something perpendicular-y about sine-y-ness that causes the cross product to get bigger the closer vectors are to perpendicular. I liked being forced to talk about dot and cross products more intuitively and spatially.
Of course, the other thing that’s going on in my vector-related life is taking representation theory and thinking about vector spaces in the abstract a lot. So much so that a vector means to me, at a most basic and intuitive level, the kind of object that lives in a vector space. Vectors are things you can add but probably not multiply, things that live over a bunch of things that you can multiply them by. It’s kind of funny to me that this one little R-3 example of a vector space is getting so much play in our physics class just because it represents the physical universe! There’s something tickling about the experience, remembering and realizing that these cartesian coordinate vectors are where all of the abstraction came from, but there’s also something disturbing about it.
I wrote last year about the things professors and textbook writers find “obvious” and “not obvioius” and sometimes I see myself starting to be convinced over to their point of view. I’m afraid of becoming so enculturated into math that my intuition, my idea of what’s “obvious” or “natural,” is permanently changed.
That isn’t perfectly put. But I want to post here more often, so I guess I have to turn some half-baked thoughts loose on the internet.
