Matrix representation for linear transformation on $\mathbb{R}^{3}$ I am trying to figure out how to solve this problem:
Find a matrix representation for the following linear transformation on $\mathbb{R}^{3}$: A clockwise rotation by $60^{\circ}$ around the $x$-axis.
The answer is:
$$
\begin{bmatrix} 1 & 0 & 0 \\ 0 & \frac{1}{2} & -\frac{\sqrt{3}}{2} \\ 0 & \frac{\sqrt{3}}{2} & \frac{1}{2} \end{bmatrix}
$$
The only thing I can deduce from my book is that the lower right $2 \times 2$ matrix is
$$
\begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}
$$
but I don't know where they got the first row or first column values from.

Thinking about the mechanical process of rotating a physical object in three dimensions, it is clear that this involves going around an axis. Then focussing on the plane perpendicular to the axis it is the usual 2d-rotation.
So a 3d-rotation means a choice of axis (line) and then quantum of rotation (when measured in radians a  number between $0$ and $2\pi$.)
