Aufgaben zur Drehung

Die Gerade $g$ mit $y=\frac{1}{4}x-1$ soll mit dem Winkel $\alpha =50°$ um das Zentrum $Z(1|2)$ gedreht werden.

Gesucht ist jetzt also die Geradengleichung von $g^{\prime }$.

Man wählt einen allgemeinen Punkt $P_n(x|\frac{1}{4}x-1)$ auf der Geraden und dreht diesen um $50°$ um $Z$.

$\overrightarrow{ZP}=\left(\begin{array}{c}x-1\\ \frac{1}{4}x-1-2\end{array}\right)=\left(\begin{array}{c}x-1\\ \frac{1}{4}x-3\end{array}\right)$

$\begin{array}{rcl}\overrightarrow{O{Q}^{\prime }}& =& \left(\begin{array}{c}\cos \alpha -\sin \alpha \\ \sin \alpha \cos \alpha \end{array}\right)\cdot \overrightarrow{ZP}\\ & & \\ & =& \left(\begin{array}{c}\cos 50°-\sin 50°\\ \sin 50°\cos 50°\end{array}\right)\cdot \left(\begin{array}{c}x-1\\ \frac{1}{4}x-3\end{array}\right)\\ & & \\ & =& \left(\begin{array}{c}\mathrm{0,64}\mathrm{0,77}\\ -\mathrm{0,77}\mathrm{0,64}\end{array}\right)\cdot \left(\begin{array}{c}x-1\\ \frac{1}{4}x-3\end{array}\right)\\ & & \\ & =& \left(\begin{array}{c}\mathrm{0,64}\cdot (x-1)+\mathrm{0,77}\cdot (\frac{1}{4}x-3)\\ -\mathrm{0,77}\cdot (x-1)+\mathrm{0,64}\cdot (\frac{1}{4}x-3)\end{array}\right)\\ & & \\ & =& \left(\begin{array}{c}\mathrm{0,83}x-\mathrm{2,95}\\ -\mathrm{0,61}x-\mathrm{1,15}\end{array}\right)\end{array}$

Jetzt muss nur noch die Parallelverschiebung durchgeführt werden:

$\begin{array}{rcl}\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right)& =& \overrightarrow{O{Q}^{\prime }}+\overrightarrow{OZ}\\ & =& \left(\begin{array}{c}\mathrm{0,83}x-\mathrm{2,95}\\ -\mathrm{0,61}x-\mathrm{1,15}\end{array}\right)+\left(\begin{array}{c}1\\ 2\end{array}\right)\\ & =& \left(\begin{array}{c}\mathrm{0,83}x-\mathrm{2,95}+1\\ -\mathrm{0,61}x-\mathrm{1,15}+2\end{array}\right)\end{array}$

$\Rightarrow P_n^{\prime }(\mathrm{0,83}x-\mathrm{1,95}|-\mathrm{0,61}x+\mathrm{0,85})$

$\begin{array}{rcl}\stackrel{⟶}{Z}& =& \left(\begin{array}{c}x-3\\ -\frac{1}{2}x+2-(-1)\end{array}\right)\\ & & \\ & =& \left(\begin{array}{c}x-3\\ -\frac{1}{2}x+3\end{array}\right)\stackrel{⟶}{O}\\ & & \\ & =& \left(\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right)\cdot \stackrel{⟶}{Z}\\ & & \\ & =& \left(\begin{array}{cc}\cos 120°& -\sin 120°\\ \sin 120°& \cos 120°\end{array}\right)\cdot \left(\begin{array}{c}x-3\\ -\frac{1}{2}x+3\end{array}\right)\\ & & \\ & =& \left(\begin{array}{cc}-\frac{1}{2}& -\frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2}& -\frac{1}{2}\end{array}\right)\cdot \left(\begin{array}{c}x-3\\ -\frac{1}{2}x+3\end{array}\right)\\ & & \\ & =& \left(\begin{array}{c}\frac{(\sqrt{3}-2)x}{4}+\frac{3(1-\sqrt{3})}{2}\\ \frac{(2\sqrt{3}+1)x}{4}-\frac{3(\sqrt{3}+1)}{2}\end{array}\right)\end{array}$

$\begin{array}{rcl}\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right)& =& \stackrel{⟶}{O}+\stackrel{⟶}{O}\\ & & \\ & =& \left(\begin{array}{c}\frac{(\sqrt{3}-2)x}{4}+\frac{3(1-\sqrt{3})}{2}\\ \frac{x(2\sqrt{3}+1)}{4}-\frac{3(\sqrt{3}+1)}{2}\end{array}\right)+\left(\begin{array}{c}3\\ -1\end{array}\right)\\ & & \\ & =& \left(\begin{array}{c}\frac{(\sqrt{3}-2)x}{4}+\frac{9-3\sqrt{3}}{2}\\ \frac{x(2\sqrt{3}+1)}{4}-\frac{3\sqrt{3}+5}{2}\end{array}\right)\end{array}$

$\Rightarrow P^{\prime }(\frac{(\sqrt{3}-2)x}{4}+\frac{9-3\sqrt{3}}{2}|\frac{x(2\sqrt{3}+1)}{4}-\frac{3\sqrt{3}+5}{2})$

$\Rightarrow \begin{array}{rclll}{x}^{\prime }& =& \frac{(\sqrt{3}-2)x}{4}+\frac{9-3\sqrt{3}}{2}& & (1)\\ y^{\prime }& =& \frac{(2\sqrt{3}+1)}{4}x-\frac{3\sqrt{3}+5}{2}& & (2)\end{array}$

Löse $(1)$ nach $x$ auf.

$\begin{array}{rclll}x& =& \frac{4}{\sqrt{3}-2}{x}^{\prime }-\frac{12(3-\sqrt{3})}{\sqrt{3}-2}& & (1^{\prime })\\ y^{\prime }& =& \frac{2\sqrt{3}+1}{4}x-\frac{3\sqrt{3}+5}{2}& & (2)\end{array}$

Setze $(1^{\prime })$ in $(2)$ ein.

$y^{\prime }=\frac{2\sqrt{3}+1}{\sqrt{3}-2}{x}^{\prime }+\frac{19(-\sqrt{3}+1)}{2(\sqrt{3}-2)}$

Das kannst du nun noch runden und erhähst die Geradengleichung für $g^{\prime }$.

$y^{\prime }=\mathrm{16,66}{x}^{\prime }+\mathrm{25,95}$

$\overrightarrow{Z{P}_n}=\left(\begin{array}{c}x-(-3)\\ -3x+2-\mathrm{0,5}\end{array}\right)$ $=\left(\begin{array}{c}x+3\\ -3x+\mathrm{1,5}\end{array}\right)$

$\begin{array}{rcl}\overrightarrow{O{Q}^{\prime }}& =& \left(\begin{array}{c}\cos \alpha -\sin \alpha \\ \sin \alpha \cos \alpha \end{array}\right)\cdot \overrightarrow{Z{P}_n}\\ & & \\ & =& \left(\begin{array}{c}\cos 45°-\sin 45°\\ \sin 45°\cos 45°\end{array}\right)\cdot \left(\begin{array}{c}x+3\\ -3x+\mathrm{1,5}\end{array}\right)\\ & & \\ & =& \left(\begin{array}{c}\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\frac{1}{\sqrt{2}}\end{array}\right)\cdot \left(\begin{array}{c}x+3\\ -3x+\mathrm{1,5}\end{array}\right)\\ & & \\ & =& \left(\begin{array}{c}\frac{4}{\sqrt{2}}x+\frac{3}{2\sqrt{2}}\\ \frac{-2}{\sqrt{2}}x+\frac{9}{2\sqrt{2}}\end{array}\right)\end{array}$

$\begin{array}{rcl}\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right)& =& \stackrel{⟶}{O}+\stackrel{⟶}{O}\\ & & \\ & =& \left(\begin{array}{c}\frac{4}{\sqrt{2}}x+\frac{3}{2\sqrt{2}}\\ \frac{-2}{\sqrt{2}}x+\frac{9}{2\sqrt{2}}\end{array}\right)+\left(\begin{array}{c}-3\\ \mathrm{0,5}\end{array}\right)\\ & & \\ & =& \left(\begin{array}{c}\frac{4}{\sqrt{2}}x+\frac{3-6\sqrt{2}}{2\sqrt{2}}\\ -\frac{2}{\sqrt{2}}x+\frac{9+\sqrt{2}}{2\sqrt{2}}\end{array}\right)\end{array}$

$\Rightarrow P^{\prime }(\frac{4}{\sqrt{2}}x+\frac{3-6\sqrt{2}}{2\sqrt{2}}|-\frac{2}{\sqrt{2}}x+\frac{9+\sqrt{2}}{2\sqrt{2}})$

$\begin{array}{rcl}\left(\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right)& =& \stackrel{⟶}{O}+\stackrel{⟶}{O}\\ & & \\ & =& \left(\begin{array}{c}\frac{4}{\sqrt{2}}x+\frac{3}{2\sqrt{2}}\\ \frac{-2}{\sqrt{2}}x+\frac{9}{2\sqrt{2}}\end{array}\right)+\left(\begin{array}{c}-3\\ \mathrm{0,5}\end{array}\right)\\ & & \\ & =& \left(\begin{array}{c}\frac{4}{\sqrt{2}}x+\frac{3-6\sqrt{2}}{2\sqrt{2}}\\ -\frac{2}{\sqrt{2}}x+\frac{9+\sqrt{2}}{2\sqrt{2}}\end{array}\right)\end{array}$

$\Rightarrow P^{\prime }(\frac{4}{\sqrt{2}}x+\frac{3-6\sqrt{2}}{2\sqrt{2}}|-\frac{2}{\sqrt{2}}x+\frac{9+\sqrt{2}}{2\sqrt{2}})$

$\Rightarrow \begin{array}{rclll}{x}^{\prime }& =& \frac{4}{\sqrt{2}}x+\frac{3-6\sqrt{2}}{2\sqrt{2}}& & (1)\\ y^{\prime }& =& -\frac{2}{\sqrt{2}}x+\frac{9+\sqrt{2}}{2\sqrt{2}}& & (2)\end{array}$

Löse $(1)$ nach $x$ auf.

$\begin{array}{rclll}x& =& \frac{\sqrt{2}}{4}{x}^{\prime }-\frac{3-6\sqrt{2}}{8}& & (1^{\prime })\\ y^{\prime }& =& -\frac{2}{\sqrt{2}}x+\frac{9+\sqrt{2}}{2\sqrt{2}}& & (2)\end{array}$

Setze $(1^{\prime })$ in $(2)$ ein.

$y^{\prime }=-\frac{1}{2}{x}^{\prime }+\frac{21\sqrt{2}-8}{8}$

Das kannst du nun noch runden und erhälst die Geradengleichung für $g^{\prime }$.

$g^{\prime }:y^{\prime }=-\mathrm{0,5}{x}^{\prime }+\mathrm{2,71}$