exc 1 final adjustements

exercise1/Theorie.tex

@@ -119,26 +119,11 @@ $\Rightarrow S \times T \neq T \times S \Rightarrow$ does not commute\\
 \begin{pmatrix} x_1 & x_2 & \cdots & x_n \\ y_1&y_2&\cdots&y_n\\ z_1&z_2&\cdots&z_n \\ \end{pmatrix} \times \begin{pmatrix} 1 \\ 1 \\ \vdots \\ 1 \\ \end{pmatrix} = 
 \begin{pmatrix}  \sum\limits_{i=1} x_i \\ \sum\limits_{i=1} y_i \\ \sum\limits_{i=1} z_i \end{pmatrix}
 \end{gather*}
+\newpage
+
 \subsection*{c)}
-\begin{align*}
-L(p) &= length(p_x - ((p_x \cdot (\vec{e} + \vec{d} \cdot t)) \cdot \vec{d} \cdot t) - \vec{e}) \\
-Vector &= ( L(p_x), L(p_{x+1}), ... , L(p_{x+n})) \hspace{6em} n \widehat{=} Anz. Punkte \\
-\end{align*}
-
-Berechnung für: \\
-$p_1 = (-1, -1, 1); p_2 = (2, 1, -2); p_3 = (2, 1, -3); p_4 = (-1, -2, 1); p_5 = (3, -1, 0)$
-\begin{align*}
-L(p_1) &= length(p_1 - ((p_1 \cdot (\vec{e} + \vec{d} \cdot t)) \cdot \vec{d} \cdot t) - \vec{e}) \\
-L(p_2) &= length(p_2 - ((p_2 \cdot (\vec{e} + \vec{d} \cdot t)) \cdot \vec{d} \cdot t) - \vec{e}) \\
-L(p_3) &= length(p_3 - ((p_3 \cdot (\vec{e} + \vec{d} \cdot t)) \cdot \vec{d} \cdot t) - \vec{e}) \\
-L(p_4) &= length(p_4 - ((p_4 \cdot (\vec{e} + \vec{d} \cdot t)) \cdot \vec{d} \cdot t) - \vec{e}) \\
-L(p_5) &= length(p_5 - ((p_5 \cdot (\vec{e} + \vec{d} \cdot t)) \cdot \vec{d} \cdot t) - \vec{e}) \\
-\\
-Vector &= ( L(p_1), L(p_2), L(p_3), L(p_4), L(p_5)) \\
-\end{align*}
-
 \begin{gather*}
-M = \begin{pmatrix} M_{11} & M_{12} & M_{13}  & t_x \\ M_{21}&M_{22}& M_{23}&t_y\\ M_{31}&M_{32}&M_{33}&t_z \\ p_x&p_y&p_z&1 \\ \end{pmatrix};
+ \tilde{M} = \begin{pmatrix} M_{11} & M_{12} & M_{13}  & t_x \\ M_{21}&M_{22}& M_{23}&t_y\\ M_{31}&M_{32}&M_{33}&t_z \\ p_x&p_y&p_z&1 \\ \end{pmatrix};
 e = \begin{pmatrix} 5 \\ 10 \\ 5 \\\end{pmatrix}= t;
 p = \begin{pmatrix} 0 \\ 0 \\ 0 \\ \end{pmatrix}
 \end{gather*}
@@ -147,7 +132,18 @@ z_\phi = \frac {\begin{pmatrix} 0\\0\\1\\ \end{pmatrix} \times p_x} {1 \times |p
 y_\phi = \frac {\begin{pmatrix} 0\\1\\0\\ \end{pmatrix} \times p_x} {1 \times |p_x|} \Rightarrow \begin{pmatrix} cos(y_\phi) & 0 & sin(y_\phi)  \\ 0 &1 &0\\ -sin(z_\phi)&0&cos(y_\phi) \\ \end{pmatrix}=R_y\\
 x_\phi = \frac {\begin{pmatrix} 1\\0\\0\\ \end{pmatrix} \times p_x} {1 \times |p_x|} \Rightarrow \begin{pmatrix} 0 & 0 & 0  \\ 0 &cos(x_\phi)&-sin(x_\phi)\\ 0&sin(x_\phi)&cos(x_\phi) \\ \end{pmatrix}=R_x\\
 M = R_x \times R_y \times R_z \\
-\Rightarrow Vector = \begin{pmatrix}
+\end{gather*}
+Berechnung für: \\
+$p_1 = (-1, -1, 1); p_2 = (2, 1, -2); p_3 = (2, 1, -3); p_4 = (-1, -2, 1); p_5 = (3, -1, 0)$
+\begin{gather*}
+\Rightarrow Vector =\begin{pmatrix}
+p_1 \times \tilde{M_1}\\
+p_2 \times \tilde{M_2}\\
+p_3 \times \tilde{M_3}\\
+p_4 \times \tilde{M_4}\\
+p_5 \times \tilde{M_5}\\ \end{pmatrix}
+=
+ \begin{pmatrix}
 6.323746006808568\\
 2.672777625063053\\
 1.6528549605644152\\