### Rotation Matrices

#### Part 2: Two-Dimensional
Rotation Matrices

- Test your result from Part
1 by rotating the standard unit vector
**e**_{1} in **R**^{2} 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
**e**_{2}.
- Compute
**B**^{6},
**B**^{12}, **B**^{18}, and **B**^{24}. 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 **B**^{2}, **K**^{4},
**K**^{2}B, **KBK**, and **BK**^{2}. 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.

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