For more info on Euler Sequences, notation and convention see the generic entry on Euler angle sequences.
The rotation matrices are
Carrying out the matrix multiplication from right to left
Finaly leaving us with the Euler 123 sequence
Testing some escape charachters for html category with a generator has an injective cogenerator" now escape ” with "
![$\displaystyle R_3(\psi) = \left[ \begin{array}{ccc} c_{\psi} & s_{\psi} & 0 \ -s_{\psi} & c_{\psi} & 0 \ 0 & 0 & 1 \end{array} \right]$](http://images.physicslibrary.org/cache/objects/42/l2h/img2.png "$\displaystyle R_3(\psi) = \left[ \begin{array}{ccc} c_{\psi} & s_{\psi} & 0 \ -s_{\psi} & c_{\psi} & 0 \ 0 & 0 & 1 \end{array} \right]$")
![$\displaystyle R_2(\theta) = \left[ \begin{array}{ccc} c_{\theta} & 0 & -s_{\theta} \ 0 & 1 & 0 \ s_{\theta} & 0 & c_{\theta} \end{array} \right]$](http://images.physicslibrary.org/cache/objects/42/l2h/img3.png "$\displaystyle R_2(\theta) = \left[ \begin{array}{ccc} c_{\theta} & 0 & -s_{\theta} \ 0 & 1 & 0 \ s_{\theta} & 0 & c_{\theta} \end{array} \right]$")
![$\displaystyle R_1(\phi) = \left[ \begin{array}{ccc} 1 & 0 & 0 \ 0 & c_{\phi} & s_{\phi} \ 0 & -s_{\phi} & c_{\phi} \end{array} \right]$](http://images.physicslibrary.org/cache/objects/42/l2h/img4.png "$\displaystyle R_1(\phi) = \left[ \begin{array}{ccc} 1 & 0 & 0 \ 0 & c_{\phi} & s_{\phi} \ 0 & -s_{\phi} & c_{\phi} \end{array} \right]$")
![$R_2(\theta)R_1(\phi) = \left[ \begin{array}{ccc} c_{\theta} & 0 & -s_{\theta} \... ... s_{\theta} & -c_{\theta} s_{\phi} & c_{\theta} c_{\phi} \end{array} \right] $](http://images.physicslibrary.org/cache/objects/42/l2h/img5.png "$R_2(\theta)R_1(\phi) = \left[ \begin{array}{ccc} c_{\theta} & 0 & -s_{\theta} \... ... s_{\theta} & -c_{\theta} s_{\phi} & c_{\theta} c_{\phi} \end{array} \right] $")
![$R_3(\psi)R_2(\theta)R_1(\phi) = \left[ \begin{array}{ccc} c_{\psi} c_{\theta} &... ... s_{\theta} & -c_{\theta} s_{\phi} & c_{\theta} c_{\phi} \end{array} \right] $](http://images.physicslibrary.org/cache/objects/42/l2h/img6.png "$R_3(\psi)R_2(\theta)R_1(\phi) = \left[ \begin{array}{ccc} c_{\psi} c_{\theta} &... ... s_{\theta} & -c_{\theta} s_{\phi} & c_{\theta} c_{\phi} \end{array} \right] $")