For more info on Euler Sequences, notation and convention see the generic entry on Euler angle sequences. \\
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The rotation matrices are
![{\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)
![{\displaystyle R_{2}(\theta )=\left[{\begin{matrix}c_{\theta }&0&-s_{\theta }\\0&1&0\\s_{\theta }&0&c_{\theta }\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd2d14d893a246b3de1a21325b8d03813e80664e)
![{\displaystyle R_{3}(\psi )=\left[{\begin{matrix}c_{\psi }&s_{\psi }&0\\-s_{\psi }&c_{\psi }&0\\0&0&1\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63e956480bacaa2f9661e0c5d7211e895c3f3dd4)
Carrying out the matrix multiplication from right to left \\
![{\displaystyle R_{2}(\theta )R_{3}(\psi )=\left[{\begin{matrix}c_{\theta }&0&-s_{\theta }\\0&1&0\\s_{\theta }&0&c_{\theta }\end{matrix}}\right]\left[{\begin{matrix}c_{\psi }&s_{\psi }&0\\-s_{\psi }&c_{\psi }&0\\0&0&1\end{matrix}}\right]=\left[{\begin{matrix}c_{\theta }c_{\psi }&c_{\theta }s_{\psi }&-s_{\theta }\\-s_{\psi }&c_{\psi }&0\\s_{\theta }c_{\psi }&s_{\theta }s_{\psi }&c_{\theta }\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80e4dc8fc5909bf5d3c1a1563f3994a3ab6c7a74)
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Finaly leaving us with the Euler 321 sequence \\
![{\displaystyle R_{1}(\phi )R_{2}(\theta )R_{3}(\psi )=\left[{\begin{matrix}c_{\theta }c_{\psi }&c_{\theta }s_{\psi }&-s_{\theta }\\-c_{\phi }s_{\psi }+s_{\phi }s_{\theta }c_{\psi }&c_{\phi }c_{\psi }+s_{\phi }s_{\theta }s_{\psi }&s_{\phi }c_{\theta }\\s_{\phi }s_{\psi }+c_{\phi }s_{\theta }c_{\psi }&-s_{\phi }c_{\psi }+c_{\phi }s_{\theta }s_{\psi }&c_{\phi }c_{\theta }\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe69e6d9456273940a54eac93a3fa41893aebfdc)
