«K12/K13» 56. Hausaufgabe «PDF», «POD»

0.0.1 ↑ 56. Hausaufgabe

0.0.1.1 ↑ Angabe einer bestimmten Strecke

A(5,3,4); \quad B(-10, -3, -8);$A\left(5,3,4\right);B\left(-10,-3,-8\right);$

\begin{array}{rcl} {} \left[AB\right] &=& {} \left\{ {} \begin{pmatrix}q_1\\q_2\\q_3\end{pmatrix} \middle| {} \begin{pmatrix}q_1\\q_2\\q_3\end{pmatrix} = {} \begin{pmatrix}5\\3\\4\end{pmatrix} + {} k \begin{pmatrix}-10 - 5\\-3 - 3\\-8 - 4\end{pmatrix} {} \text{mit } k \in \left[0, 1\right] {} \right\} = \vspace*{0.8em}\\ {} &=& {} \left\{ {} \begin{pmatrix}q_1\\q_2\\q_3\end{pmatrix} \middle| {} \begin{pmatrix}q_1\\q_2\\q_3\end{pmatrix} = {} \begin{pmatrix}-10\\-3\\-8\end{pmatrix} + {} k \begin{pmatrix}5 - \left(-10\right)\\3 - \left(-3\right)\\4 - \left(-8\right)\end{pmatrix} {} \text{mit } k \in \left[0, 1\right] {} \right\} = \vspace*{0.8em}\\ {} &=& {} \left\{ {} \begin{pmatrix}q_1\\q_2\\q_3\end{pmatrix} \middle| {} \begin{pmatrix}q_1\\q_2\\q_3\end{pmatrix} = {} \vec A + k \overrightarrow{AB} {} \text{ mit } k \in \left[0, 1\right] {} \right\} = \vspace*{0.8em}\\ {} &=& {} \left\{ {} \begin{pmatrix}q_1\\q_2\\q_3\end{pmatrix} \middle| {} \begin{pmatrix}q_1\\q_2\\q_3\end{pmatrix} = {} \vec B + k \overrightarrow{BA} {} \text{ mit } k \in \left[0, 1\right] {} \right\}\!; \end{array}$\begin{array}{ccc}\hfill \left[AB\right]& \hfill =\hfill & \left\{\left(\begin{array}{c}\hfill q_1\hfill \\ \hfill q_2\hfill \\ \hfill q_3\hfill \end{array}\right)\mid \left(\begin{array}{c}\hfill q_1\hfill \\ \hfill q_2\hfill \\ \hfill q_3\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 5\hfill \\ \hfill 3\hfill \\ \hfill 4\hfill \end{array}\right)+k\left(\begin{array}{c}\hfill -10-5\hfill \\ \hfill -3-3\hfill \\ \hfill -8-4\hfill \end{array}\right)\text{mit }k\in \left[0,1\right]\right\}=\hfill \\ \hfill & \hfill =\hfill & \left\{\left(\begin{array}{c}\hfill q_1\hfill \\ \hfill q_2\hfill \\ \hfill q_3\hfill \end{array}\right)\mid \left(\begin{array}{c}\hfill q_1\hfill \\ \hfill q_2\hfill \\ \hfill q_3\hfill \end{array}\right)=\left(\begin{array}{c}\hfill -10\hfill \\ \hfill -3\hfill \\ \hfill -8\hfill \end{array}\right)+k\left(\begin{array}{c}\hfill 5-\left(-10\right)\hfill \\ \hfill 3-\left(-3\right)\hfill \\ \hfill 4-\left(-8\right)\hfill \end{array}\right)\text{mit }k\in \left[0,1\right]\right\}=\hfill \\ \hfill & \hfill =\hfill & \left\{\left(\begin{array}{c}\hfill q_1\hfill \\ \hfill q_2\hfill \\ \hfill q_3\hfill \end{array}\right)\mid \left(\begin{array}{c}\hfill q_1\hfill \\ \hfill q_2\hfill \\ \hfill q_3\hfill \end{array}\right)=\overrightarrow{A}+k\overrightarrow{AB}\text{ mit }k\in \left[0,1\right]\right\}=\hfill \\ \hfill & \hfill =\hfill & \left\{\left(\begin{array}{c}\hfill q_1\hfill \\ \hfill q_2\hfill \\ \hfill q_3\hfill \end{array}\right)\mid \left(\begin{array}{c}\hfill q_1\hfill \\ \hfill q_2\hfill \\ \hfill q_3\hfill \end{array}\right)=\overrightarrow{B}+k\overrightarrow{BA}\text{ mit }k\in \left[0,1\right]\right\};\hfill \end{array}$