So I was teaching the cross product in my multivariable calc class on Friday. For those who are a bit rusty, is the notation for the cross product between 3-dimensional vectors and . It is defined as:
Prompted by a question from a student, I was struck with two related questions. One is: why does yield a vector orthogonal to both and ? That question doesn’t mean I want a proof: I can do that. But I want some intuitive sense of why it works.
The second question is: can we find cross products in other coordinate systems, in the same way? How generalizable is this whole “take the determinant of a 3×3 matrix”? Would finding a cross product using a determinant work in, say, spherical or cylindrical coordinate — where instead of putting , , as the top row of the matrix, you put the fundamental unit “direction” vectors in these other coordinate systems? I haven’t played around with these coordinate systems in a long time, but I suspect the answer is no. (And to see that, I’d just have to do a simple test case, which wouldn’t work out… and that ought to be good enough.) I have to run to a picnic now, so maybe I’ll try this out later.
Hm, this looks like it could be turned into a good question for the next problem set, or even a good project for the multivariable students.
This train of thought also got me musing on the “what are the origins of vector analysis?” question. I haven’t had time to do any serious research, but it appears that one important book on this question is A History of Vector Analysis by Michael Crowe.
Matrices rarely fail to confound me when I get outside the bounds of what I have to teach the kids how to do. I recently had my mind blown by observing the transformations in 2-space you get by multiplying matrices by other matrices of various determinants.
Why is the cross product orthogonal? Dunno! Let’s see, the determinant of a 2×2 is the area of the parallelogram defined by the two vectors, the determinant of a 3×3 is the volume of the parallelpiped. So….with cross product you’re using two 3-d vectors and (i, j, k)…yeah. Not seeing it. Let us know what you come up with.