Let's look at each given angle of rotation and match it with the corresponding resulting vector matrix.
1. [tex]\(\frac{\pi}{2}\)[/tex] radians:
- The result is [tex]\(\begin{bmatrix} 5 \\ -3 \end{bmatrix}\)[/tex]
2. [tex]\(\frac{3\pi}{4}\)[/tex] radians:
- The result is [tex]\(\begin{bmatrix} 5.65685425 \\ 1.41421356 \end{bmatrix}\)[/tex]
3. [tex]\(\frac{5\pi}{3}\)[/tex] radians:
- The result is [tex]\(\begin{bmatrix} -5.83012702 \\ 0.09807621 \end{bmatrix}\)[/tex]
4. [tex]\(\frac{5\pi}{6}\)[/tex] radians:
- The result is [tex]\(\begin{bmatrix} 5.09807621 \\ 2.83012702 \end{bmatrix}\)[/tex]
5. [tex]\(\frac{3\pi}{2}\)[/tex] radians:
- The result is [tex]\(\begin{bmatrix} -5 \\ 3 \end{bmatrix}\)[/tex]
6. [tex]\(\frac{4\pi}{3}\)[/tex] radians:
- The result is [tex]\(\begin{bmatrix} -2.83012702 \\ 5.09807621 \end{bmatrix}\)[/tex]
Here are the pairs:
- [tex]\(\frac{\pi}{2}\)[/tex] matches with [tex]\(\begin{bmatrix} 5 \\ -3 \end{bmatrix}\)[/tex]
- [tex]\(\frac{3\pi}{4}\)[/tex] matches with [tex]\(\begin{bmatrix} 5.65685425 \\ 1.41421356 \end{bmatrix}\)[/tex]
- [tex]\(\frac{5\pi}{3}\)[/tex] matches with [tex]\(\begin{bmatrix} -5.83012702 \\ 0.09807621 \end{bmatrix}\)[/tex]
- [tex]\(\frac{5\pi}{6}\)[/tex] matches with [tex]\(\begin{bmatrix} 5.09807621 \\ 2.83012702 \end{bmatrix}\)[/tex]
- [tex]\(\frac{3\pi}{2}\)[/tex] matches with [tex]\(\begin{bmatrix} -5 \\ 3 \end{bmatrix}\)[/tex]
- [tex]\(\frac{4\pi}{3}\)[/tex] matches with [tex]\(\begin{bmatrix} -2.83012702 \\ 5.09807621 \end{bmatrix}\)[/tex]
