### Rotation Matrices

#### Part 3: Three-Dimensional Rotation Matrices

In R3 a vector can be rotated about any one of the three axes. The 3-dimensional versions of the rotation matrix A are the following matrices: P rotates a vector in R3 about the x3-axis, Q about the x1-axis, and R about the x2-axis. These are not the only possible rotations in 3-space, of course, but we will limit our attention in this module to these possibilities.

1. What feature of each of the matrices P, Q, R tells us quickly the axis about which the rotation is being done?
2. We saw earlier that multiplication of 2-dimensional rotation matrices is commutative, even though matrix multiplication in general is not commutative. We'll try this for 3-dimensional rotation matrices. Let P30 and P45 be the matrices for rotations of 30 and 45 degrees, respectively, around the x3-axis. Compute P30P45 and P45P30. What do you observe?
3. Now compute P30R45 and R45P30. What do you observe? Hold an object (such as your textbook) in front of you, and rotate it as indicated in these product matrices. Try to convince yourself that what you observed mathematically is consistent with reality!
4. Try to generalize what you computed in the preceding steps. In particular, are any 3-dimensional rotation matrices multiplicatively commutative? If so, which ones?
5. Suppose an image is stored in computer memory as a set of coordinates in 3-dimensional space. Assume that when the object is displayed on the view screen, the x1-axis is perpendicular to the screen, the x2-axis is horizontal, and the x3-axis is vertical. Thus, the x2-x3 plane is on the surface of the screen. The software can perform rotations by multiplying each point (as a vector) by an appropriate rotation matrix and then displaying the result. If we want to display the object so that it is first flipped over from our right to our left, and then the axis projected toward us is tilted upward 20 degrees, what matrix can the computer use to do this transformation? Explain your reasoning, and compute the matrix in your worksheet. (A sample object -- a plane in 3-space -- is defined for plotting in your worksheet. You may experiment with this plot in order to visualize the transformation in this step.)

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