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author | Andreas Müller <andreas.mueller@ost.ch> | 2021-06-14 07:26:10 +0200 |
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committer | GitHub <noreply@github.com> | 2021-06-14 07:26:10 +0200 |
commit | 114633b43a0f1ebedbc5dfd85f75ede9841f26fd (patch) | |
tree | 18e61c7d69883a1c9b69098b7d36856abaed5c1e /vorlesungen/slides/7/quaternionen.tex | |
parent | Delete buch.pdf (diff) | |
parent | Fix references.bib (diff) | |
download | SeminarMatrizen-114633b43a0f1ebedbc5dfd85f75ede9841f26fd.tar.gz SeminarMatrizen-114633b43a0f1ebedbc5dfd85f75ede9841f26fd.zip |
Merge branch 'master' into master
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-rw-r--r-- | vorlesungen/slides/7/quaternionen.tex | 74 |
1 files changed, 74 insertions, 0 deletions
diff --git a/vorlesungen/slides/7/quaternionen.tex b/vorlesungen/slides/7/quaternionen.tex new file mode 100644 index 0000000..f526366 --- /dev/null +++ b/vorlesungen/slides/7/quaternionen.tex @@ -0,0 +1,74 @@ +% +% quaternionen.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Quaternionen} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Quaternionen} +$4$-dimensionaler $\mathbb{R}$-Vektorraum +\[ +\mathbb{H} += +\langle 1,i,j,k\rangle_{\mathbb{R}} +\] +mit Rechenregeln +\[ +i^2=j^2=k^2=ijk=-1 +\] +$x=x_0+x_1i+x_2j+x_3k\in\mathbb{H}$ +\begin{itemize} +\item<2-> Realteil: $\operatorname{Re}x=x_0$ +\item<3-> Vektorteil: $\operatorname{Im}x=x_1i+x_2j+x_3k$ +\item<4-> Konjugation: $\overline{x}=\operatorname{Re}x-\operatorname{Im}x$ +\item<5-> Norm: $|x|^2 = x\overline{x} = x_0^2+x_1^2+x_2^2+x_3^2$ +\item<6-> Inverse: $x^{1}= \overline{x}/x\overline{x}$ +\end{itemize} +\end{block} +\end{column} +\begin{column}{0.50\textwidth} +\uncover<7->{% +\begin{block}{Skalarprodukt und Vektorprodukt} +\begin{align*} +pq +&= +\operatorname{Re}p \operatorname{Re}q +- +\operatorname{Im}p\cdot \operatorname{Im}q +\\ +&\phantom{=} ++ +\operatorname{Re}p\operatorname{Im}q ++ +\operatorname{Im}p\operatorname{Re}q ++ +\operatorname{Im}p\times\operatorname{Im}q +\end{align*} +\end{block}} +\uncover<8->{% +\begin{block}{Einheitsquaternionen} +$q\in \mathbb{H}$, $|q|=1, q^{-1}=\overline{q}$ +\end{block}} +\uncover<9->{% +\begin{block}{Polardarstellung} +\[ +q = \cos\frac{\alpha}2 + \vec{n} \sin\frac{\alpha}2 +\] +\vspace{-8pt} +\begin{itemize} +\item<10-> +Drehmatrix: 9 Parameter, 6 Bedingungen +\item<11-> +Quaternionen: 4 Parameter, 1 Bedingung +\end{itemize} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup |