The rotation matrix around the Y-axis by an angle $\theta$ is given by:

$$R_y(\beta) = \begin{pmatrix} \cos(\beta) & 0 & \sin(\beta) \\ 0 & 1 & 0 \\ -\sin(\beta) & 0 & \cos(\beta) \end{pmatrix}$$

The general expression of the transformation matrix is :

$$R_y(\beta) = \begin{pmatrix} cos(\beta) && 0 && sin(\beta) && 0 \\ 0 && 1 && 0 && 0 \\ -sin(\beta) && 0 && cos(\beta) && 0 \\ 0 && 0 && 0 && 1 \end{pmatrix}$$

This matrix is used to rotate a point or a vector in three-dimensional space around the Y-axis.

Example

Suppose we want to rotate the point $P(1, 0, 0)$ by an angle of $90^\circ$ (or $\frac{\pi}{2}$ radians) around the Y-axis.

1. Define the rotation angle:
   $$\theta = 90^\circ = \frac{\pi}{2}$$

2. Calculate the rotation matrix:

$$R_y\left(\frac{\pi}{2}\right) = \begin{pmatrix} \cos\left(\frac{\pi}{2}\right) & 0 & \sin\left(\frac{\pi}{2}\right) \\ 0 & 1 & 0 \\ -\sin\left(\frac{\pi}{2}\right) & 0 & cos\left(\frac{\pi}{2}\right) \end{pmatrix} = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ -1 & 0 & 0 \end{pmatrix}$$

3. Apply the rotation matrix to the point: Let $P = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$.

   Multiply the rotation matrix by the point:

$$P' = R_y\left(\frac{\pi}{2}\right) \cdot P = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ -1 & 0 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ -1 \end{pmatrix}$$

Result

After rotating the point $P(1, 0, 0)$ by $90^\circ$ around the Y-axis, the new coordinates of the point become $P'(0, 0, -1)$.