Hallo bh,
\(\overrightarrow{AM}\) = \(\begin{pmatrix} 5-3 \\ -3 - (-5) \end{pmatrix}\) = \(\begin{pmatrix} 2 \\ 2 \end{pmatrix}\) , dazu senkrecht steht \(\vec{n}\) = \(\begin{pmatrix} -1 \\ 1 \end{pmatrix}\)
\(\overrightarrow{OC}\) = \(\overrightarrow{OA}\) + 2 * \(\overrightarrow{AM}\) = \(\begin{pmatrix} 3 + 2 * 2 \\ -5 + 2 * 2 \end{pmatrix}\) = \(\begin{pmatrix} 7 \\ -1 \end{pmatrix}\) → C(7 | - 1)
\(\overrightarrow{BM}\) = \(\overrightarrow{MD}\) sind senkrecht zu \(\overrightarrow{AM}\) , also || \(\vec{n}\)
und haben die Länge (den Betrag) 1/2*√8.
\(\overrightarrow{OD}\) = \(\overrightarrow{OM}\) + 1/2 * √8 / |\(\vec{n}\)| * \(\vec{n}\) = \(\overrightarrow{OM}\) + 1/2 * √8 / √2 * \(\vec{n}\)
= \(\overrightarrow{OM}\) + \(\vec{n}\) = \(\begin{pmatrix} 5 - 1 \\ -3 + 1 \end{pmatrix}\) = \(\begin{pmatrix} 4 \\ -2 \end{pmatrix}\) → D(4 | - 2)
Wegen \(\overrightarrow{AB}\) || \(\overrightarrow{DC}\) gilt:
\(\overrightarrow{OB}\) = \(\overrightarrow{OA}\) + \(\overrightarrow{DC}\) = \(\begin{pmatrix} 3 + 7 - 4 \\ -5 - 1 + 2 \end{pmatrix}\) = \(\begin{pmatrix} 6 \\ -4 \end{pmatrix}\) → B(6 | - 4)
Gruß Wolfgang
