Let us denote the axes after the rotational motion as $x_1^{\prime }$, $x_2^{\prime }$, and $x_3^{\prime }$. In this case, the configuration of the axes resulting from a ${45}^{\circ }$ rotation about the $x_2$ axis can be represented as shown in Figure-1. As observed in Figure-1, since the rotation is performed about the $x_2$ axis, the $x_2$ axis remains unchanged; therefore, the $x_2^{\prime }$ axis coincides with the original $x_2$ axis in direction. On the other hand, as a result of the rotation, the $x_1$ axis shifts into the region $x_1>0$ and within the $x_1\text{–}{x}_3$ plane, and is denoted by $x_1^{\prime }$. Similarly, the $x_3$ axis moves into the region $x_1>0$ and $x_3>0$ within the same plane, and is represented by $x_3^{\prime }$.

As seen in Figure~1, the $x_1^{\prime }$ axis lies in the $x_1\text{–}{x}_3$ plane within the region $x_1>0$ and . Therefore, the angle between $x_1^{\prime }$ and the $x_2$ axis is ${90}^{\circ }$, while the angle between $x_1^{\prime }$ and the $x_1$ axis is ${45}^{\circ }$ (see Figure-1). Using these geometric relations, the $x_1^{\prime }$ axis can be expressed in terms of $x_1$, $x_2$, and $x_3$ as $x_1^{\prime }=x_1\cos {45}^{\circ }+x_2\cos {90}^{\circ }-x_3\sin {45}^{\circ }$ Similarly, since the coordinate system is rotated about the $x_2$ axis, the $x_2$ axis remains unchanged. Consequently, the angle between $x_2^{\prime }$ and the $x_2$ axis is $0^{\circ }$, whereas the angles between $x_2^{\prime }$ and the $x_1$ and $x_3$ axes are ${90}^{\circ }$. Accordingly, the $x_2^{\prime }$ axis can be written as $x_2^{\prime }=x_1\cos {90}^{\circ }+x_2\cos 0^{\circ }+x_3\cos {90}^{\circ }$ Applying the same reasoning to the $x_3^{\prime }$ axis, it is observed that $x_3^{\prime }$ lies in the $x_1\text{–}{x}_3$ plane within the region $x_1>0$ and $x_3>0$. Hence, the angle between $x_3^{\prime }$ and the $x_2$ axis is ${90}^{\circ }$, while the angle between $x_3^{\prime }$ and the $x_3$ axis is ${45}^{\circ }$ (see Figure~1). Using these geometric relations, the $x_3^{\prime }$ axis can be expressed as $x_3^{\prime }=x_1\sin {45}^{\circ }+x_2\cos {90}^{\circ }+x_3\cos {45}^{\circ }$

$x_1^{\prime }=x_1\cos {45}^{\circ }+x_2\cos {90}^{\circ }-x_3\sin {45}^{\circ }$ $x_2^{\prime }=x_1\cos {90}^{\circ }+x_2\cos 0^{\circ }+x_3\cos {90}^{\circ }$ $x_3^{\prime }=x_1\sin {45}^{\circ }+x_2\cos {90}^{\circ }+x_3\cos {45}^{\circ }$ Since $\sin {45}^{\circ }=\cos {45}^{\circ }$, these transformation equations can be written in the following simplified form: $x_1^{\prime }=x_1\cos {45}^{\circ }+x_2\cos {90}^{\circ }-x_3\cos {45}^{\circ }$ $x_2^{\prime }=x_1\cos {90}^{\circ }+x_2\cos 0^{\circ }+x_3\cos {90}^{\circ }$ $x_3^{\prime }=x_1\cos {45}^{\circ }+x_2\cos {90}^{\circ }+x_3\cos {45}^{\circ }$

Using $\cos {45}^{\circ }=\frac{1}{\sqrt{2}}$ and $\cos {90}^{\circ }=0$ in the matrix equation, we obtain