Chapter 1, Problem 1.1P

To determine

The transformation matrix for rotation about $x_2$ axis.

Answer to Problem 1.1P

The transformation matrix for rotation about $x_2$ axis is $\left(\begin{array}{ccc}\frac{1}{\sqrt{2}}& 0& -\frac{1}{\sqrt{2}}\\ 0& 1& 0\\ \frac{1}{\sqrt{2}}& 0& \frac{1}{\sqrt{2}}\end{array}\right)$.

Explanation of Solution

Consider the rotation given in the below figure.

Rotation matrix for rotation about $x_2$ or y axis in 3-D is,

    $R_y\left(\theta \right)=\left(\begin{array}{ccc}\cos \theta & 0& -\sin \theta \\ 0& 1& 0\\ \sin \theta & 0& \cos \theta \end{array}\right)$

Here, $R_y\left(\theta \right)$ is the transformation matrix and $\theta$ is the angle of rotation.

Conclusion:

Substitute $45°$ for $\theta$ to get $R_y\left(\theta \right)$.

    $\begin{array}{c}{R}_y\left(\theta \right)=\left(\begin{array}{ccc}\cos 45°& 0& -\sin 45°\\ 0& 1& 0\\ \sin 45°& 0& \cos 45°\end{array}\right)\\ =\left(\begin{array}{ccc}\frac{1}{\sqrt{2}}& 0& -\frac{1}{\sqrt{2}}\\ 0& 1& 0\\ \frac{1}{\sqrt{2}}& 0& \frac{1}{\sqrt{2}}\end{array}\right)\end{array}$

Therefore, transformation matrix for rotation about $x_2$ axis is $\left(\begin{array}{ccc}\frac{1}{\sqrt{2}}& 0& -\frac{1}{\sqrt{2}}\\ 0& 1& 0\\ \frac{1}{\sqrt{2}}& 0& \frac{1}{\sqrt{2}}\end{array}\right)$.