For more info on Euler Sequences, notation and convention see the generic entry on Euler angle sequences.
R 121 ( ϕ , θ , ψ ) = R 1 ( ψ ) R 2 ( θ ) R 1 ( ϕ ) {\displaystyle R_{121}(\phi ,\theta ,\psi )=R_{1}(\psi )R_{2}(\theta )R_{1}(\phi )}
The rotation matrices are
R 1 ( ψ ) = [ 1 0 0 0 c ψ s ψ 0 − s ψ c ψ ] , R 2 ( θ ) = [ c θ 0 − s θ 0 1 0 s θ 0 c θ ] , R 1 ( ϕ ) = [ 1 0 0 0 c ϕ s ϕ 0 − s ϕ c ϕ ] . {\displaystyle R_{1}(\psi )=\left[{\begin{matrix}1&0&0\\0&c_{\psi }&s_{\psi }\\0&-s_{\psi }&c_{\psi }\end{matrix}}\right],\qquad R_{2}(\theta )=\left[{\begin{matrix}c_{\theta }&0&-s_{\theta }\\0&1&0\\s_{\theta }&0&c_{\theta }\end{matrix}}\right],\qquad R_{1}(\phi )=\left[{\begin{matrix}1&0&0\\0&c_{\phi }&s_{\phi }\\0&-s_{\phi }&c_{\phi }\end{matrix}}\right].}
Carrying out the matrix multiplication from right to left
R 2 ( θ ) R 1 ( ϕ ) = [ c θ 0 − s θ 0 1 0 s θ 0 c θ ] [ 1 0 0 0 c ϕ s ϕ 0 − s ϕ c ϕ ] = [ c θ s θ s ϕ − s θ c ϕ 0 c ϕ s ϕ s θ − c θ s ϕ c θ c ϕ ] {\displaystyle R_{2}(\theta )R_{1}(\phi )=\left[{\begin{matrix}c_{\theta }&0&-s_{\theta }\\0&1&0\\s_{\theta }&0&c_{\theta }\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 }s_{\phi }&-s_{\theta }c_{\phi }\\0&c_{\phi }&s_{\phi }\\s_{\theta }&-c_{\theta }s_{\phi }&c_{\theta }c_{\phi }\end{matrix}}\right]}
Finally leaving us with the Euler 121 sequence
R 1 ( ψ ) R 2 ( θ ) R 1 ( ϕ ) = [ c θ s θ s ϕ − s θ c ϕ s ψ s θ c ψ c ϕ − s ψ c θ s ϕ c ψ s ϕ + s ψ c θ c ϕ s θ c ψ − s ψ c ϕ − c ψ c θ s ϕ − s ψ s ϕ + c ψ c θ c ϕ ] {\displaystyle R_{1}(\psi )R_{2}(\theta )R_{1}(\phi )=\left[{\begin{matrix}c_{\theta }&s_{\theta }s_{\phi }&-s_{\theta }c_{\phi }\\s_{\psi }s_{\theta }&c_{\psi }c_{\phi }-s_{\psi }c_{\theta }s_{\phi }&c_{\psi }s_{\phi }+s_{\psi }c_{\theta }c_{\phi }\\s_{\theta }c_{\psi }&-s_{\psi }c_{\phi }-c_{\psi }c_{\theta }s_{\phi }&-s_{\psi }s_{\phi }+c_{\psi }c_{\theta }c_{\phi }\end{matrix}}\right]}
![{\displaystyle R_{1}(\psi )=\left[{\begin{matrix}1&0&0\\0&c_{\psi }&s_{\psi }\\0&-s_{\psi }&c_{\psi }\end{matrix}}\right],\qquad R_{2}(\theta )=\left[{\begin{matrix}c_{\theta }&0&-s_{\theta }\\0&1&0\\s_{\theta }&0&c_{\theta }\end{matrix}}\right],\qquad 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/c90e53d081a6199ba629fb213716e55285087cdd)
![{\displaystyle R_{2}(\theta )R_{1}(\phi )=\left[{\begin{matrix}c_{\theta }&0&-s_{\theta }\\0&1&0\\s_{\theta }&0&c_{\theta }\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 }s_{\phi }&-s_{\theta }c_{\phi }\\0&c_{\phi }&s_{\phi }\\s_{\theta }&-c_{\theta }s_{\phi }&c_{\theta }c_{\phi }\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdd34645c0119afceb4171b92248188c067be766)
![{\displaystyle R_{1}(\psi )R_{2}(\theta )R_{1}(\phi )=\left[{\begin{matrix}c_{\theta }&s_{\theta }s_{\phi }&-s_{\theta }c_{\phi }\\s_{\psi }s_{\theta }&c_{\psi }c_{\phi }-s_{\psi }c_{\theta }s_{\phi }&c_{\psi }s_{\phi }+s_{\psi }c_{\theta }c_{\phi }\\s_{\theta }c_{\psi }&-s_{\psi }c_{\phi }-c_{\psi }c_{\theta }s_{\phi }&-s_{\psi }s_{\phi }+c_{\psi }c_{\theta }c_{\phi }\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78d6ae5363abef8ef6df73ba318f707fee05b4a0)
