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 \\
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Finaly leaving us with the Euler 321 sequence \\