Aufgaben zur Parallelverschiebung

Addiere den Vektor $\stackrel{\rightharpoonup }{v}$ zu dem Ortsvektor $\stackrel{\rightharpoonup }{P}$

$\stackrel{\rightharpoonup }{P^{\prime }}=\stackrel{\rightharpoonup }{P}+\stackrel{\rightharpoonup }{v}=\left(\begin{array}{c}-2\\ 1\end{array}\right)+\left(\begin{array}{c}1\\ -3\end{array}\right)$

$x_{P^{\prime }}=-2+1=-1$

$y_{P^{\prime }}=1+(-3)=-2$

$\to P^{\prime }(-1|-2)$

$\stackrel{\rightharpoonup }{P^{\prime }}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\cdot \stackrel{\rightharpoonup }{P}+\stackrel{\rightharpoonup }{v}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\cdot \left(\begin{array}{c}-2\\ 1\end{array}\right)+\left(\begin{array}{c}1\\ -3\end{array}\right)$

Führe die Matrix-Vektor-Multiplikation durch.

$\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\cdot \left(\begin{array}{c}-2\\ 1\end{array}\right)=\left(\begin{array}{c}-2\\ 1\end{array}\right)+\left(\begin{array}{c}1\\ -3\end{array}\right)$

Addiere die Vektoren

$\left(\begin{array}{c}-2\\ 1\end{array}\right)+\left(\begin{array}{c}1\\ -3\end{array}\right)=\left(\begin{array}{c}-1\\ -2\end{array}\right)$

$\to \stackrel{\rightharpoonup }{P^{\prime }}=\left(\begin{array}{c}-1\\ -2\end{array}\right)$

$\stackrel{\rightharpoonup }{P^{\prime }}=\stackrel{\rightharpoonup }{P}+\stackrel{\rightharpoonup }{v}=\left(\begin{array}{c}-\mathrm{3,2}\\ \mathrm{2,4}\end{array}\right)+\left(\begin{array}{c}\mathrm{1,4}\\ -\mathrm{1,7}\end{array}\right)$

$x_{P^{\prime }}=-\mathrm{3,2}+\mathrm{1,4}=-\mathrm{1,8}$

$y_{P^{\prime }}=\mathrm{2,4}+(-\mathrm{1,7})=\mathrm{0,7}$

$\to P^{\prime }(-\mathrm{1,8}|\mathrm{0,7})$

$\stackrel{\rightharpoonup }{P^{\prime }}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\cdot \stackrel{\rightharpoonup }{P}+\stackrel{\rightharpoonup }{v}=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\cdot \left(\begin{array}{c}-\mathrm{3,2}\\ \mathrm{2,4}\end{array}\right)+\left(\begin{array}{c}\mathrm{1,4}\\ -\mathrm{1,7}\end{array}\right)$

Führe die Matrix-Vektor-Multiplikation durch

$\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\cdot \left(\begin{array}{c}-\mathrm{3,2}\\ \mathrm{2,4}\end{array}\right)+\left(\begin{array}{c}\mathrm{1,4}\\ -\mathrm{1,7}\end{array}\right)=\left(\begin{array}{c}-\mathrm{3,2}\\ \mathrm{2,4}\end{array}\right)+\left(\begin{array}{c}\mathrm{1,4}\\ -\mathrm{1,7}\end{array}\right)$

Addiere die Vektoren

$\left(\begin{array}{c}-\mathrm{3,2}\\ \mathrm{2,4}\end{array}\right)+\left(\begin{array}{c}\mathrm{1,4}\\ -\mathrm{1,7}\end{array}\right)=\left(\begin{array}{c}-\mathrm{1,8}\\ \mathrm{0,7}\end{array}\right)$

$\to \stackrel{\rightharpoonup }{P^{\prime }}=\left(\begin{array}{c}-\mathrm{1,8}\\ \mathrm{0,7}\end{array}\right)$