The three points P, Q and R from T1:

I'll use little cubes to represent these points.

You'll recall the parameterization of a plane from class. This can be quite handy for representing pieces of a plane. For example, we can show the parallelogram generated by the vectors PQ and PR, anchored at P:

(You might recall that I asked you to find a way to generate just the triangle. You do that yet?)

I want a disk centered at P in the plane containing the points. To help with this I'll first find a pair of orthogonal unit vectors in the plane. Here's are some utility functions, whose utility should be clear:

Given two nonparallel vectors and , you can easily show that the vector

is orthogonal to . This means that we have the following two orthogonal unit vectors to play with, using PQ and PR in place of and :

The magnitudes of the vectors PQ and PR are

so a disk with radius 5 will show things nicely:

Keep playing. The cross product can be useful to get us out of the plane of the triangle.

Okay, enough of that. (Unless you can think of something you'd like to see...)

Here's a picture that corresponds to the extra-credit problem on T1. We want to rotate the point R about the line , through an angle of .  The trick is to use some convenient orthogonal unit vectors. But we already have them, because is a unit vector along PQ, is in the plane of PQR and is perpendicular to , and is a unit vector perpendicular to both and . From what we once discussed in class, the point should be

,

where is the distance from R to and S is the point on nearest to R, namely,

.

Let's have R run circles around S.

The answer to the T1 problem is as follows.

Refer to our course web page to see some nice "Live3D" versions of these figures.