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A rotation matrix is fundamental when it comes to transforming vectors in 3D space. Specifically, when you want to rotate a vector around one of the coordinate axes (x, y or z), you use a rotation matrix. In this task, we are rotating about the z-axis. For counterclockwise rotation by an angle \( \theta \), the matrix is: \[ R = \begin{pmatrix} \cos \theta & -\sin \theta & 0 \ \sin \theta & \cos \theta & 0 \ 0 & 0 & 1 \end{pmatrix} \]
Plugging in \( \theta = 30^{\circ} = \frac{\pi}{6} \), we get: \[ R = \begin{pmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} & 0 \ \frac{1}{2} & \frac{\sqrt{3}}{2} & 0 \ 0 & 0 & 1 \end{pmatrix} \]
This rotation matrix will rotate any vector in \( \mathbb{R}^3 \) 30 degrees counterclockwise around the z-axis.

The reflection matrix is used to flip a vector over a specific plane. In this exercise, the transformation involves reflecting through the xy-plane. The reflection can be captured by this 3x3 matrix: \[ F = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & -1 \end{pmatrix} \]
This matrix leaves the x and y components unchanged, but it multiplies the z component by -1, effectively flipping any vector over the xy-plane.

To combine multiple transformations into one, we use matrix multiplication. Here, we need to multiply the reflection matrix by the rotation matrix to get our final transformation: \[ T = F \cdot R = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & -1 \end{pmatrix} \cdot \begin{pmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} & 0 \ \frac{1}{2} & \frac{\sqrt{3}}{2} & 0 \ 0 & 0 & 1 \end{pmatrix} \]
Carrying out the multiplication step-by-step:

• First row calculation: \( 1 \cdot \frac{\sqrt{3}}{2} + 0 \cdot \frac{1}{2} + 0 \cdot 0 = \frac{\sqrt{3}}{2} \).
• Second row calculation: \( 1 \cdot -\frac{1}{2} + 0 \cdot \frac{\sqrt{3}}{2} + 0 \cdot 0 = -\frac{1}{2} \).
• Third row calculation: \(0 \cdot \frac{\sqrt{3}}{2} + 0 \cdot \frac{1}{2} + -1 \cdot 1 = -1 \).

Thus, the final transformation matrix is:
\[ T = \begin{pmatrix} \frac{\sqrt{3}}{2} & -\frac{1}{2} & 0 \ \frac{1}{2} & \frac{\sqrt{3}}{2} & 0 \ 0 & 0 & -1 \end{pmatrix} \]
This matrix first rotates vectors 30 degrees counterclockwise about the z-axis and then reflects them through the xy-plane.