Vectors 'n' such

I've spent my idle time this evening putting on paper some ideas for learning activities...for the last few days I've been stalled at this point in the planning process, trying to come up with ideas for engaging classroom exercises. I've been spinning my wheels, I think because I've been looking at it the wrong way: I've been trying to create, ex nihilo, "brilliant" activities that will enrapture the students, and then trying to jam those squarish pedagogical pegs into round holes by struggling to find the foundational material those activities might embody.

Then, while I was lying on the couch in between chapters of my leisure reading (ah, summer!), I just thought for a few minutes about vectors: where do they come from? Where do we see them? What's their natural habitat look like?

My first thought was of the RGB vectors which describe colors in computer displays. They're vectors, after all. And it's not difficult to tell Mathematica to paint a unit cube full of colored pixels which vividly demonstrate the meaning of these vectory incarnations. RGB analysis lends colorful (ha ha) insight into linear combinations and orthogonality.

Then I thought of balanced chemical equations...every student's going to have some experience in that, right? And viewed in the right way, we can learn a lot about linear combinations, linear independence, subspaces, and so forth, just by pulling apart molecular formulas.

Once they've done a little work with these applications, I can set them off on an exploratory analysis of the usual representation of vectors in the plane.

We'll see if I'll have time tomorrow to refine these activities, and coordinate them with the aspects of "vector" with which I want to begin the semester.