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\cfrac{b}{a} $ & $\varphi = arctan \cfrac{b}{a}$ \\ $a < 0$ & $\varphi = \arctan \cfrac{b}{a} + 180^\circ $ & $\varphi = arctan \cfrac{b}{a} + \pi $ \\ @@ -381,6 +406,13 @@ $a = 0, b > 0 $ & $\varphi = 90^\circ $ & $\varphi = \cfrac{\pi}{2} $ $a = 0, b < 0 $ & $\varphi = 270^\circ $ & $\varphi = \cfrac{3}{2}\pi $ \\ $a = 0, b = 0 $ & $\varphi = 0^\circ $ & $\varphi = 0 $ \\ \end{tablebox} + +\end{sectionbox} + +\begin{sectionbox} + + + \subsection{Potenzen von i} @@ -438,9 +470,6 @@ $a = 0, b = 0 $ & $\varphi = 0^\circ $ & $\varphi = 0 $ \\ \end{align*} Es gibt immer $n$ Ergebnisse die in ${ z }_{ k } $ für $k= 0$ bis $k= n-1$ berechnet werden. -\end{sectionbox} - -\begin{sectionbox} \subsection{Formen} \textbf{Kartesische Form:} \begin{align*} @@ -455,6 +484,156 @@ Es gibt immer $n$ Ergebnisse die in ${ z }_{ k } $ für $k= 0$ bis $k= n-1$ bere \end{align*} \end{sectionbox} +% Vektoren und Matritzen +% ---------------------------------------------------------------------- +\section{Vektoren und Matritzen} + +\begin{sectionbox} +Matritzen vom Typ (m,1) sind Vektoren (1-Spaltig). Die Zeilen eines Vektors sind auch die Dimension des Vektors. Ein Zeilen Vektor ist eine Matritze vom Typ (1, n). + +\subsection{Rechenoperationen} +\subsubsection{Skalar} +Die Addition, Subtraktion und Multiplikation von Matritzen mit einem Skalar (einer Zahl) c. +\begin{align*} +-1 \cdot A &= -A \\ +c \cdot A &= A \cdot c = Ac = cA \\ +{c}_{1} \cdot \left( {c}_{2} \cdot A \right) &= \left( {c}_{1} \cdot {c}_{2} \right) \cdot A \\\left({c}_{1} + {c}_{2} \right) \cdot A &= {c}_{1} \cdot A + {c}_{2} \cdot A \\ +c \cdot \left(A + B \right) &= cA + cB +\end{align*} + +\subsubsection{Multiplikation} +\begin{itemize} +\item Zeile von Matrix A mal Spalte von Matrix B +\item Matrix A muss so viele Spalten haben wie Matrix B Zeilen hat +\item Nicht Kommutativ!! $A \cdot B \neq B \cdot A$ +\end{itemize} + +\subsubsection{Determinante} +\begin{itemize} +\item Ist $\det A \neq 0$ dann ist die Matrix invertierbar +\item Ist $\det A = 0$ dann ist die Matrix Linear abhängig +\item Schachbrettmuster (beginnend oben links mit + - + -..) +\item Entwicklung am einfachsten nach der Spalte oder Zeile mit den meisten 0er. +\end{itemize} + +\begin{align*} +A &= \begin{pmatrix} 0 & 1 & 3 \\ 4 & 2 & 0 \\ 0 & 1 & 5 \end{pmatrix} \\ +det A &= -4 \cdot \begin{vmatrix} 1 & 3 \\ 1 & 5 \end{vmatrix} \\ +&= -4 \cdot \left(1*5 - 1*3\right) = -8 +\end{align*} + +\subsection{Inverse Matrix} +Invertierbar sind nur Matritzen des Typ (n, n) also quadratische Matritzen. +Eine Matrix ist dann Invertierbar wenn die Determinante $ \neq $ 0 ergibt. +\begin{tablebox}{ll} +${A}^{-1} \cdot A = I$ & ${\left( A \cdot B \right)}^{-1} = {B}^{-1} \cdot {A}^{-1}$ \\ +$A \cdot {A}^{-1} = I$ & $I \cdot A = A $ \\ +$I \cdot {A}^{-1} = {A}^{-1}$ & \\ +\end{tablebox} + +\end{sectionbox} + + +% Vektoren +% ---------------------------------------------------------------------- +\section{Vektoren} + +\begin{sectionbox} + +\subsection{Vektor Aufstellen} + +\begin{minipage}{0.49\textwidth} +\begin{align*} + A &= \begin{pmatrix} {a}_{1} & {a}_{2} & {a}_{3} \end{pmatrix} \\ + B &= \begin{pmatrix} {b}_{1} & {b}_{2} & {b}_{3} \end{pmatrix} +\end{align*} + +\end{minipage} +\begin{minipage}{0.49\textwidth} +\begin{align*} +\overrightarrow{AB} = \begin{pmatrix} +{b}_{1} - {a}_{1} \\ +{b}_{2} - {a}_{2} \\ +{b}_{3} - {a}_{3} \\ +\end{pmatrix} +\end{align*} +\end{minipage} + +\subsection{Rechenoperation} +\begin{tablebox}{ll} +Addition & $ \overrightarrow{a} = \begin{pmatrix} 1 \\ 2 \end{pmatrix} + \begin{pmatrix} {b}_{1} \\ {b}_{2}\end{pmatrix} = \begin{pmatrix} 1 + {b}_{1} \\ 2 + {b}_{2} \end{pmatrix} $ \\ +Multiplikation & $ \overrightarrow{a} = c \cdot \begin{pmatrix} {b}_{1} \\ {b}_{2} \end{pmatrix} = \begin{pmatrix} c {b}_{1} \\ c {b}_{2} \end{pmatrix} $ \\ +Betrag & $ \vert \overrightarrow{a} \vert = \sqrt{{{b}_{1}}^{2} + {{b}_{1}}^{2} } $ \\ +Skalarpordukt & $\overrightarrow{a} \ast \overrightarrow{b} = {a}_{1}{b}_{1} + {a}_{2}{b}_{2} + {a}_{3}{b}_{3} $ \\ +Kreuzprodukt (Abb. \ref{kreuzprodukt})& $\overrightarrow{a} \times \overrightarrow{b} = \overrightarrow{n} = \begin{pmatrix} +{a}_{2}{b}_{3} - {a}_{3}{b}_{2} \\ +{a}_{3}{b}_{1} - {a}_{1}{b}_{3} \\ +{a}_{1}{b}_{2} - {a}_{2}{b}_{1} \\ +\end{pmatrix} $ \\ +\end{tablebox} +Ist das Skalarprodukt = 0 dann sind die Vektoren orthogonal (senkrecht) zueinander! + + +\end{sectionbox} + +% Geraden und Ebenen +% ---------------------------------------------------------------------- +\section{Geraden und Ebenen} + +\begin{sectionbox} +\subsection{Schnittpunkte} +\begin{minipage}{0.49\textwidth} +\textbf{Gerade} +Den Schnittpunkt von zwei geraden erhält man indem man die beiden Geradengleichungen gleich setzt. +\end{minipage} +\begin{minipage}{0.49\textwidth} +\textbf{Ebene} +Bei Einer Ebene funktioniert die Berechnung des Schnittpunktes analog zu dem einer Geraden. +\end{minipage} + +\subsection{Winkel} +\begin{minipage}{0.49\textwidth} +\textbf{Gerade und Gerade} +\begin{align*} +\cos \alpha = \left| \frac{\xrightarrow{a} \cdot \xrightarrow{b} }{\vert \xrightarrow{a} \vert \cdot \vert \xrightarrow{b} \vert} \right| +\end{align*} +\end{minipage} +\begin{minipage}{0.49\textwidth} +\textbf{Gerade und Ebene} +\begin{align*} +\cos \alpha = \left| \frac{\xrightarrow{a} \cdot \xrightarrow{b} }{\vert \xrightarrow{a} \vert \cdot \vert \xrightarrow{b} \vert} \right| +\end{align*} +\end{minipage} + +\subsection{Formen} +\subsubsection{Geraden} +Allgemeine Form: +\begin{align*} +\overrightarrow{x} = \overrightarrow{p} + t \cdot \overrightarrow{u} +\end{align*} +$\overrightarrow{p}$ = Stützvektor und $\overrightarrow{u}$ = Richtungsvektor. + +\subsubsection{Ebenen} +\begin{tablebox}{ll} +Parameterform & +$E:\overrightarrow{x} = \overrightarrow{p} + t \cdot \overrightarrow{u} + s \cdot \overrightarrow{v} $ \\ +Normalform & $\overrightarrow{n} \cdot \left( \overrightarrow{x} - \overrightarrow{p} \right) = 0 $\\ +\end{tablebox} +$\overrightarrow{p}$ = Stützvektor und $\overrightarrow{u}$,$\overrightarrow{v}$ = Spannvektor + +\textbf{Umformen:} +\begin{enumerate} + \item Parameter $\rightarrow$ Normalform + \item $\overrightarrow{n} = \overrightarrow{u} \times \overrightarrow{v}$ (Kreuzprodukt der Richtungsvektoren + \item $\overrightarrow{n} \cdot \left( \overrightarrow{x} - \overrightarrow{p} \right) = 0 $ + \item Koordinatenform aufstellen \\ + $\overrightarrow{n} \cdot \overrightarrow{x} = \overrightarrow{n} \cdot \overrightarrow{p}$ + (Normalform aus multipliziert) +\end{enumerate} + + +\end{sectionbox} + % ====================================================================== % End % ======================================================================