How to find the matrix given the rotation operation?

Question

For example, Y is a $3 \times 3$ matrix representing a counterclockwise rotation of $\frac{\pi}{4}$ radians about the y-axis (assuming the positive axis is pointing towards you).

Answer 1

When rotating a vector about the $y$-axis its tip traces out an arc in a plane parallel to the $xz$-plane, that is, its $x$- and $z$-coordinate change but its $y$-coordinate is unchanged. The corresponding rotation matrix is

$$ R_{y,\theta} = \left[ \begin{array}{ccc} \cos\theta & 0 & -\sin\theta\\ 0 & 1 & 0\\ \sin\theta & 0 & \cos\theta \end{array} \right] $$