Mathematics 41, section 1 -- Analysis 3

Discussion 1 -- Vector Algebra and Geometry

January 19, 1998

Discussion Days

To learn mathematics well, it pays to be an active participant figuring out things either individually or with others, not just a passive spectator. For this reason, about once a week this semester, we will be breaking the class down into three- or four-person groups for discussion days to work on discussion questions like the ones below on this sheet. Many of you have probably worked this way before in mathematics or other classes. If so, then you will probably find that the group work in this class will be similar to what you have seen before. If not, then you may have to adjust your thinking about how you do mathematics, because this style of working can be quite different from a standard lecture/homework/exams class. Here are some guidelines and things to think about:

1. A group work best when you think of it as a true joint effort -- a collaboration. The results of the discussion days will be one set of group solutions to the questions and I will be giving a single group grade to encourage this.
2. The goal is for everyone to understand all of the work the group does and contribute to it. But different people will inevitably be at different places in their understanding of the mathematics at different times, so that contribution will take different forms. If you think you do see something, be prepared to explain it to the others in your group. There is no better way to make sure you really understand something than to try to explain it to someone else! On the other hand, if you don't see what one of your groupmates is getting at, don't be shy about asking questions. Just formulating a specific question can often start the process of developing understanding. By keeping people honest you might also keep the group as a whole from going off on an unproductive tangent!
3. One of the other goals of this classroom approach is to help you develop interpersonal skills that will be very useful in the work teams common in lots of today's jobs. Some of you may not find this aspect of what we do to be easy, especially if you are used to thinking of classwork as a competition between you and the other students. There may be times when you feel your group has become disfunctional because of personality conflicts or other disagreements. If that happens, give it a "cooling-off period". If that doesn't help, feel free to talk to me about the group issues, either as a whole group or individually. I plan to keep the same group assignments for about half the semester, then reshuffle the class. So you will have an opportunity to get to know your groupmates pretty well, but there will be a chance for a change if that becomes necessary.

Mathematical Background For Today

Recall that last Friday, we saw that vectors in the plane or space can be added (in geometric form, via the parallelogram law), and multiplied by scalars. Moreover, if we have a collection of vectors and place all tails at the origin, then the set of heads will describe some set of points.

Discussion Questions

A) Describe in words the set of points in the plane we get in this way if v = (1,2), u = (-3,1), and we consider the sets of vectors:
1) S₁ = {u + c v : c in R }.
2) S₂ = {u + c v : c > 0}.
3) S₃ = {u + c v : c in [0,1]}.

B) Same question for u = (3,1,0), v = (1,2,0), w = (2,1,-1), and
1) S₁ = {c u + d v : c,d in R}.
2) S₂ = {c u + d v : c,d > 0}.
3) S₃ = {c u + d v : c,d in [0,1]}.
4) S₄ = { w + c u + d v : c,d in R }.
5) S₅ = { w + c u + d v : c,d > 0}.
6) S₆ = { w + c u + d v : c,d in [0,1]}.
7) S₇ = {e w + c u + d v : c,d,e in [0,1]}.

C) The set of points in S₇ in question B is called a parallelotope. Taking {e w + c u + d v : c,d,e in [0,1]} for three general vectors u, v, w will "usually" give something similar. Are there special cases too, though, for which you get something different? For instance, what happens if u = (1,1,1), v = (-1,0,2) and w = (0,1,3)? Explain.

Assignment

Group writeups due in class, Wednesday, January 21.