Part 2: Two-Dimensional
- Test your result from Part
1 by rotating the standard unit vector e1 in R2 through an angle of 60 degrees. First, use some scratch paper and trigonometry
to see what the answer should be. Enter your hand-calculated image vector
in the worksheet. Then let B be the 2 x 2 rotation matrix for an angle
of 60 degrees. (Enter B as a specific instance of A for the given
angle. Recall that the built-in trigonometric functions expect angles in
radians.) Check that your matrix B gives the same result as the hand calculation.
- Repeat the preceding step
with the standard unit vector e2.
- Compute B6,
B12, B18, and B24. Explain your results
in terms of rotations. In particular how could you have predicted the particular
matrices that the computer algebra system produced?
- Let K be the 2 x 2 rotation
matrix for a rotation of 15 degrees. Compute B2, K4,
K2B, KBK, and BK2. Compare the results and explain
what you see.
- In general we know that
matrix multiplication is not commutative; i.e., if B and K are both n x n matrices, then usually BK is not the same as KB. However, rotation matrices
are a special case. Explain why, for any two 2-dimensional rotation matrices
Q and R, it must follow that QR = RQ. Then use B and K from the preceding
step to illustrate your argument.
- Explain why the product
of any two 2 x 2 rotation matrices is another rotation matrix. Illustrate
your argument with a specific example.
modules at math.duke.edu