>[!example] Expressing 2D [[Rotation Transformation|rotation]] as a [[Change of Basis Matrix|change of basis]] in $\R^2$: >$\huge \begin{align} > >{}_{\alpha}P_{\beta} &= \mat{ [u_{1}]_{\alpha} & [u_{2}]_{\alpha}} \\ > >\alpha &= \braket{\hat i, \hat j} \\ >\beta &= \braket{\hat i', \hat j'} \\ >\hat i ' &= \hat i \cos \theta + \hat j \sin \theta \\ >\hat j ' &= -\hat j \sin \theta + \hat j \cos \theta \\ > >{}_\alpha P_{\beta} &= \mat{ \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta } \\ > >{}_{\alpha}P_{\beta} &={}_{\alpha}P_{\beta}^{-1} > >\end{align}$ ### Rotations in $\R^3$ as [[Change of Basis for Transformations|Change of Basis]] Let $\mathcal Z_{\phi}$ denote precesion around the around the $z$-axis by $\phi$, such that $z=z'$ $\huge \begin{matrix} \alpha = \set{ \hat i, \hat j, \hat k } & \beta = \set{ \hat i', \hat j', \hat k' } \end{matrix} \begin{align}\\ \end{align}$ $\huge \begin{align} \begin{cases} \hat i' = \hat i \cos \phi + \hat j \sin \phi \\ \hat j' = -\hat i \sin \phi + \hat j \cos \phi \\ \hat k' = \hat k \\ \end{cases} \end{align}$ $\huge \begin{align} P_{\beta \to \alpha} &= \mat{ \cos \phi & -\sin \phi & 0 \\ \sin \phi & \cos \phi & 0 \\ 0 & 0 & 1 } \\ \pa{P_{\alpha \to \beta}}^{-1} &= \pa{P_{\beta \to \alpha}}^{\intercal} & \\&= \mat{ \cos \phi & \sin \phi & 0 \\ -\sin \phi & \cos \phi & 0 \\ 0 & 0 & 1 } \\ &= \mathcal Z_{\phi} \end{align}$ Which leads us to the definition $\huge \mathcal Z_{\phi} = \mat{ \cos \phi & \sin \phi & 0 \\ -\sin \phi & \cos \phi & 0 \\ 0 & 0 & 1 } \\ $ We can define $\mathcal X_{\theta}$ as another [[Euler Angles|Euler Rotation]], which is to say that $\mathcal X_{\theta}$ is rotation about the $x$-axis by $\theta$, this can be derived with similar reasoning as $\mathcal Z_{\phi}$. $\huge \begin{align} \mathcal X_{\theta} &= \mat{ 1 & 0 & 0 \\ 0 & \cos \theta & \sin \theta \\ 0 & -\sin \theta & \cos \theta } \end{align}$ We can define $Z_{\psi}$ which is rotation around the $z$-axis with an angle of $\psi$. $\huge \begin{align} Z_{\psi} &= \mat{ \cos \psi & \sin \psi & 0 \\ -\sin \psi & \cos \psi & 0 \\ 0 & 0 & 1 } \end{align} $ These terms together form Euler Angles $\huge (\phi,\theta,\psi) = \text{Euler Angles} $ You can use this system to perform general rotations with euler angles. $\huge \begin{align} P_{\alpha \to \delta} &= P_{\gamma \to \delta} P_{\beta \to \gamma} P_{\alpha \to \beta} \\ P_{\alpha \to \delta} &= \mathcal Z_{\psi} \mathcal X_{\theta} \mathcal Z_{\phi} \end{align} $ We could also add another rotation about the $y$-axis by an angle of $\beta$, to account for some faults / [[Gimble Lock]] using euler angles. $\huge \mathcal Y_{\beta} = \mat{ \cos \beta &0& -\sin \beta \\ 0&1&0 \\ \sin \beta &0& \cos \beta } $ Which is used in [[Tait-Bryan Angles]].