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\begin{document}


\title{Quaternion division}

\maketitle
we will define the division of quaternions as 
\[
\frac{q}{p}=qp^{-1}=q\frac{\overline{p}}{\left| p\right| ^{2}}\]
 Given \( m=\left| p\right| ^{2}=p_{s}^{2}+p_{a}^{2}+p_{b}^{2}+p_{c}^{2} \),
we can write \( \frac{q}{p} \)component wise as follows:
\[
\begin{array}{cl}
s & =\frac{q_{s}p_{s}+q_{a}p_{a}+q_{b}p_{b}+q_{c}p_{c}}{m}\\
a & =\frac{q_{a}p_{s}-q_{s}p_{a}-q_{b}p_{c}+q_{c}p_{b}}{m}\\
b & =\frac{q_{b}p_{s}-q_{s}p_{b}-q_{c}p_{a}+q_{a}p_{c}}{m}\\
c & =\frac{q_{c}p_{s}-q_{s}p_{c}-q_{a}p_{b}+q_{b}p_{a}}{m}
\end{array}\]


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