QUESION4
Local Rotation
Choose one $\cdot 5$ points
Write the local rotation matrix of 30 about the local Y-axis, ${ }^{B_{R_{Y}}(30)}$
$\left(\begin{array}{ccc}\frac{\sqrt{3}}{2} & 0 & -\frac{1}{2} \\ 0 & 1 & 0 \\ \frac{1}{2} & 0 & \frac{\sqrt{3}}{2}\end{array}\right)$
$\left(\begin{array}{ccc}\frac{\sqrt{3}}{2} & 0 & \frac{1}{2} \\ 0 & 1 & 0 \\ -\frac{1}{2} & 0 & \frac{\sqrt{8}}{2}\end{array}\right)$
$\left(\begin{array}{ccc}\frac{\sqrt{3}}{2} & 0 & \frac{1}{2} \\ 0 & 1 & 0 \\ \frac{1}{2} & 0 & \frac{\sqrt{3}}{2}\end{array}\right)$

Step 1 ：The problem is asking for the local rotation matrix of 30 degrees about the local Y-axis. The rotation matrix for a rotation of θ degrees about the Y-axis is given by: \[ \begin{bmatrix} cos(\theta) & 0 & sin(\theta) \\ 0 & 1 & 0 \\ -sin(\theta) & 0 & cos(\theta) \end{bmatrix} \]

Step 2 ：We need to substitute θ = 30 degrees into this matrix. However, the trigonometric functions take arguments in radians, not degrees. So we first need to convert 30 degrees to radians. The conversion gives us θ = 0.5235987755982988 radians.

Step 3 ：Substituting θ into the matrix, we get: \[ \begin{bmatrix} cos(0.5235987755982988) & 0 & sin(0.5235987755982988) \\ 0 & 1 & 0 \\ -sin(0.5235987755982988) & 0 & cos(0.5235987755982988) \end{bmatrix} \]

Step 4 ：Evaluating the trigonometric functions, we get: \[ \begin{bmatrix} 0.8660254037844387 & 0 & 0.49999999999999994 \\ 0 & 1 & 0 \\ -0.49999999999999994 & 0 & 0.8660254037844387 \end{bmatrix} \]

Step 5 ：The rotation matrix we calculated matches the first option given in the question. The other two options are incorrect because they have incorrect elements in the third column of the matrix.

Step 6 ：Final Answer: The local rotation matrix of 30 degrees about the local Y-axis is \[ \boxed{ \begin{bmatrix} \frac{\sqrt{3}}{2} & 0 & \frac{1}{2} \\ 0 & 1 & 0 \\ -\frac{1}{2} & 0 & \frac{\sqrt{3}}{2} \end{bmatrix} } \]