Lösungen zu den Aufgaben 4

4.1

a) $\overrightarrow{AB}=\left(\begin{array}{c}-5+8\\ 7-5\\ -11-12\end{array}\right)=\left(\begin{array}{c}3\\ 2\\ -23\end{array}\right)$
b) $\overrightarrow{AB}=\left(\begin{array}{c}-8\\ 3\\ -11\end{array}\right)$

4.2

a) $\overrightarrow{P{P}^{\prime }}=\overrightarrow{v}=\left(\begin{array}{c}3\\ 2\\ -4\end{array}\right)$
b) $\overrightarrow{A^{\prime }}=\overrightarrow{OA}+\overrightarrow{P{P}^{\prime }}=\left(\begin{array}{c}6\\ 0\\ 0\end{array}\right)$
$\overrightarrow{B^{\prime }}=\overrightarrow{OB}+\overrightarrow{P{P}^{\prime }}=\left(\begin{array}{c}6,5\\ 4,5\\ -9\end{array}\right)$

4.3

$\overrightarrow{v_1}=\left(\begin{array}{c}5\\ 3\\ 2\end{array}\right)$

$\overrightarrow{v_2}=\left(\begin{array}{c}-4\\ 2\\ 4\end{array}\right)$

a) $\overrightarrow{O{P}^{\prime }}=\overrightarrow{OP}+\overrightarrow{v_1}\left(\begin{array}{c}2\\ 0\\ 5\end{array}\right)$

$\overrightarrow{O{P}^{″}}=\overrightarrow{O{P}^{\prime }}+\overrightarrow{v_2}\left(\begin{array}{c}-2\\ 2\\ 9\end{array}\right)$

b) $\overrightarrow{v}=\overrightarrow{v_1}+\overrightarrow{v_2}\left(\begin{array}{c}1\\ 5\\ 6\end{array}\right)=\overrightarrow{P{P}^{\prime }}$