Any [[Unit Quaternion|unit]] [[Quaternions|quaternion]] $q$ can be represented as:
$\huge \begin{align}
q = \cos \pa{\frac{\theta}{2}} + \hat n \sin \pa{\frac{\theta}{2}}
\end{align}$
Where $\hat n$ denotes a [[Unit Quaternion|Unit]] [[Quaternion Vector]].
>[!info]- Proof
>$\large i= \hat{i},j=\hat{j},k=\hat{k} $
>$\large\begin{align}
>
>q &= \cos \pa{\frac{\theta}{2}} + (b\hat{i}+c\hat{j}+d\hat{k})\sin\pa{\frac{\theta}{2}} \\
>
>q^{*} &=\cos \pa{\frac{\theta}{2}} - (b\hat{i}+c\hat{j}+d\hat{k})\sin\pa{\frac{\theta}{2}} \\
>
>
>qq^{*} &=\cos^{2}\pa{\frac{\theta}{2}} + \sin^{2}\pa{\frac{\theta}{2}}\pa{b^{2}+c^{2}+d^{2}} \\
>
>qq^{*} &=\cos^{2}\pa{\frac{\theta}{2}} + \sin^{2}\pa{\frac{\theta}{2}}{|\hat{n}|} \\
>
>qq^{*} &= \cos^{2}\left( \frac{\theta}{2} \right) + \sin^{2}\left( \frac{\theta}{2} \right)
>
>\end{align}$
A rotation as a [[Linear Transformation]] about an [[Axis]] ([[Unit Vector]]) $\hat{n}$ by an angle of $\theta$ can be written as:
$\huge \begin{align}
L(\vec v) &= q\vec v q^{*}
\end{align}$
Where $q$ is defined as the previous spelling:
$\huge \begin{align}
q = \cos \pa{\frac{\theta}{2}} + \hat n \sin \pa{\frac{\theta}{2}}
\end{align}$
>[!warning]
>Rotate $v=\braket{1,2,1}$ about $\hat n= \frac{1}{\sqrt{ 5 }} \braket{1,2,0}$. and $\theta= \frac{\pi}{3}$.
>
>$\huge \begin{align}
>q &= \cos \frac{\pi}{6} + \frac{1}{\sqrt{ 5 }}(1i+2j) \sin \frac{\pi}{6} \\
>q^{*} &= \cos \frac{\pi}{6} - \frac{1}{\sqrt{ 5 }}(1i+2j) \sin \frac{\pi}{6} \\
>
>q &= \frac{\sqrt{ 3 }}{2} + \frac{1}{2}\pa{ \frac{1}{\sqrt{ 5 }}i + \frac{2}{\sqrt{ 5 }}j }\\
>
>q^{*} &= \frac{\sqrt{ 3 }}{2} - \frac{1}{2}\pa{ \frac{1}{\sqrt{ 5 }}i + \frac{2}{\sqrt{ 5 }}j }
>\end{align}$
>
>
>$\begin{align}
>
>L(\vec v) &= q\vec v q^{*} \\
>
>&=
>
>\pa{\frac{\sqrt{ 3 }}{2} + \frac{1}{2}\pa{ \frac{1}{\sqrt{ 5 }}i + \frac{2}{\sqrt{ 5 }}j }}
>(i+2j+k)
>\pa{\frac{\sqrt{ 3 }}{2} - \frac{1}{2}\pa{ \frac{1}{\sqrt{ 5 }}i + \frac{2}{\sqrt{ 5 }}j }} \\
>&= \frac{\sqrt{ 3 }}{2}(i+2j+k) + \frac{1}{2}\pa{ \frac{1}{\sqrt{ 5 }}i^{2}
>+ \frac{2}{\sqrt{ 5 }}ij + \frac{1}{\sqrt{ 5 }}k
>+ \frac{2}{\sqrt{ 5 }}ji
>+ \frac{k}{\sqrt{ 5 }}j^{2}
>+ \frac{2}{\sqrt{ 5 }} jk
>}
>q^{*} \\
>
>&= \pa{
>
>\frac{\sqrt{ 3 }}{2}i + \sqrt{ 3 }j + \frac{\sqrt{ 3 }}{2}k
>- \frac{1}{2\sqrt{ 5 }}k
>+ \frac{1}{\sqrt{ 5 }}
>- \frac{1}{\sqrt{ 5 }}j - \frac{1}{\sqrt{ 5 }}k- \frac{2}{\sqrt{ 5 }i}
>+\frac{1}{\sqrt{ 5 }}
>}q^{*} \\
>&= \pa{
>\pa{
>
>\pa{ -\frac{1}{2\sqrt{ 5 }} - \frac{2}{\sqrt{ 5 }} }
>+ \frac{\sqrt{ 3 }}{2}- \frac{1}{\sqrt{ 5 }} }i
>+ \pa{ \sqrt{ 3 } - \frac{1}{2\sqrt{ 5 }} }j
>+ \frac{\sqrt{ 3 }}{2}k
>}
>\pa{
>
>\frac{\sqrt{ 3 }}{2}
>- \frac{1}{2\sqrt{ 5 }}i - \frac{1}{\sqrt{ 5 }}j
>
>} \cdots \\
>&= \text{who even knows anymore bro, give up and use a calculator 🥀}
>\end{align}$
>
>[!info] Relation to [[Rodrigues Formula]]
>$\huge \begin{align}
>
>L(\vec v) &=
>\pa{\cos \frac{\theta}{2}+ \hat{n} \sin \frac{\theta}{2}}\vec v \pa{
>\cos \frac{\theta}{2} - \hat n \sin \frac{\theta}{2}
>} \\
>
>&=\vec v \cos^{2} \frac{\theta}{2} -
>\vec v \hat n \sin \frac{\theta}{2} \cos \frac{\theta}{2} +
>\hat{n} \vec v^{2} \sin \frac{\theta}{2} \cos \frac{\theta}{2}
>-
>\hat{n}^{2}\vec v\sin^{2 } \frac{\theta}{2} \\
>
>&=
>
>
>\pa{
>\cos^{2} \frac{\theta}{2} - \sin ^{2} \frac{\theta}{2}
>}\vec v
>+ 2\pa{\hat{n} \times \vec v} \sin \frac{\theta}{2} \cos \frac{\theta}{2}
>\\
>&= \vec v \cos \theta + \pa{\hat{n} \times \vec v} \sin \theta - \pa{\cdots}\pa{
>\hat{n} \times \hat{n} \times \vec v
>}
>\end{align}$