For more info on Euler Sequences, notation and convention see the generic entry on Euler angle sequences. \\
\\
The rotation matrices are
![{\displaystyle R_{2}(\psi )=\left[{\begin{matrix}c_{\psi }&0&-s_{\psi }\\0&1&0\\s_{\psi }&0&c_{\psi }\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/028d20c50911cb091ec521688cb8afc3fab2a78c)
![{\displaystyle R_{3}(\theta )=\left[{\begin{matrix}c_{\theta }&s_{\theta }&0\\-s_{\theta }&c_{\theta }&0\\0&0&1\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34a0f690b735ddc78c55e84eeac41660b787fdb0)
![{\displaystyle R_{1}(\phi )=\left[{\begin{matrix}1&0&0\\0&c_{\phi }&s_{\phi }\\0&-s_{\phi }&c_{\phi }\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/192ac3559eaf3aee6d189ef8923fa25721b238a0)
Carrying out the matrix multiplication from right to left \\
![{\displaystyle R_{3}(\theta )R_{1}(\phi )=\left[{\begin{matrix}c_{\theta }&s_{\theta }&0\\-s_{\theta }&c_{\theta }&0\\0&0&1\end{matrix}}\right]\left[{\begin{matrix}1&0&0\\0&c_{\phi }&s_{\phi }\\0&-s_{\phi }&c_{\phi }\end{matrix}}\right]=\left[{\begin{matrix}c_{\theta }&s_{\theta }c_{\phi }&s_{\theta }s_{\phi }\\-s_{\theta }&c_{\theta }c_{\phi }&c_{\theta }s_{\phi }\\0&-s_{\phi }&c_{\phi }\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f18acfde6afc6f0be444b19d40488b4bee262037)
\\
Finaly leaving us with the Euler 132 sequence \\
![{\displaystyle R_{2}(\psi )R_{3}(\theta )R_{1}(\phi )=\left[{\begin{matrix}c_{\psi }c_{\theta }&c_{\psi }s_{\theta }c_{\phi }+s_{\psi }s_{\phi }&c_{\psi }s_{\theta }s_{\phi }-s_{\psi }c_{\phi }\\-s_{\theta }&c_{\theta }c_{\phi }&c_{\theta }s_{\phi }\\s_{\psi }c_{\theta }&s_{\psi }s_{\theta }c_{\phi }-c_{\psi }s_{\phi }&s_{\psi }s_{\theta }s_{\phi }+c_{\psi }c_{\phi }\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e00b27dc4d85eb7f6da6f7b15c68665a1ed0c99)
