It is known that the group $\text{SO}(3)$ of rotation-matrices (matrices $A$ with $\det(A)=1$)
are generated from three parameters. This can be expressed by the fact, that
any rotation matrix is a composition of axis rotations
$$
\begin{pmatrix}
\cos(\phi)&-\sin(\phi)&0\\
\sin(\phi)&\cos(\phi)&0\\
0&0&1\\
\end{pmatrix},
\begin{pmatrix}
\cos(\phi)&0&\sin(\phi)\\
0&1&0\\
-\sin(\phi)&0&\cos(\phi)\\
\end{pmatrix},
\begin{pmatrix}
1&0&0\\
0&\cos(\phi)&-\sin(\phi)\\
0&\sin(\phi)&\cos(\phi)\\
\end{pmatrix}
$$
The question is: Why is the second matrix (Usually called rotation around the $y$-axis ) in almost any textbook written like this?
Related to the other two matrices, I would say that the negative
$$
\begin{pmatrix}
\cos(\phi)&0&-\sin(\phi)\\
0&1&0\\
\sin(\phi)&0&\cos(\phi)\\
\end{pmatrix},
$$
is conceptual more straight forward. Any help or guidance will be appreciated.
Best Answer
Note that all the matrices listed will rotate vectors by the angle $\phi$ around the $x,y$ and $z$ axis respectively. The alternating signs is a result of the right hand screw rule. Let \begin{equation*} A=\bigl(\begin{smallmatrix} \cos(\phi) & 0 & \sin(\phi) \\ 0 & 1 & 0 \\ -\sin(\phi) & 0 & \cos(\theta) \end{smallmatrix}\bigr). \end{equation*}
Note that to be a rotation matrix, we must have $A^T=A^{-1}$ and $\det A=1$ which you can check holds by an elementary computation. The locations of all the elements in the $y-$axis rotation matrix are placed so that we have a rotation around the $y-$axis. For example, suppose we are in $\mathbb{R}^3$ and we want to rotate the vector $(0,0,1)$ (aligned with the $z-$axis) $90^o$s. Then multiplying $A$ evaluated at $\phi=90$ by this unit vector gives $(1,0,0)$ which geometrically is a $90^o$ anticlockwise direction around the $y-$axis.