Einschub: Analytische Geometrie (Wiederholung); 559/2; 382/15

Mathe_13_1.tex

@@ -1694,3 +1694,73 @@ \subsection{Wurzelfunktionen}
   f(x) = \sqrt{x^2 - 4} \qquad D = \{x \in \mathbb{R} \colon x \leq -2 \vee x \geq 2\} \\
   g(x) = \sqrt{4 - x^2} \qquad D = [-2; 2]
 \end{gather*}
+\newpage
+\section*{Einschub: Analytische Geometrie (Wiederholung)}
+\begin{exercise}{559/2}
+  \begin{gather*}
+    A(1|6|0) \qquad B(4|7|2) \qquad C(2|7|1) \qquad D(2|0|2)
+  \end{gather*}
+  \item [a]
+  \begin{gather*}
+    \vv{AB} = \begin{pmatrix}3 \\ 1 \\ 2\end{pmatrix} \qquad \vv{AC} = \begin{pmatrix}1 \\ 1 \\ 1\end{pmatrix} \qquad E \colon \vv{x} = \begin{pmatrix}1 \\ 6 \\ 0\end{pmatrix} + r \cdot \begin{pmatrix}3 \\ 1 \\ 2\end{pmatrix} + s \cdot \begin{pmatrix}1 \\ 1 \\ 1\end{pmatrix} \\
+    \vv{n} = \vv{AB} \times \vv{AC} = \begin{pmatrix}-1 \\ -1 \\ 2\end{pmatrix} \qquad E \colon \left(\vv{x} - \begin{pmatrix}1 \\ 6 \\ 0\end{pmatrix}\right) \cdot \begin{pmatrix}-1 \\ -1 \\ 2\end{pmatrix} = 0 \\\\
+    E \colon \left(\begin{pmatrix}x_1 \\ x_2 \\ x_3\end{pmatrix} - \begin{pmatrix}1 \\ 6 \\ 0\end{pmatrix}\right) \cdot \begin{pmatrix}-1 \\ -1 \\ 2\end{pmatrix} = -x_1 - x_2 + 2x_3 + 7 = 0 \\
+    -2 - 0 + 2 \cdot 2 + 7 = 9 \neq 0 \quad\Rightarrow\quad D \not\in E
+  \end{gather*}
+  \item [b]
+  \begin{gather*}
+    g \colon \vv{x} = \begin{pmatrix}2 \\ 0 \\ 2\end{pmatrix} + r \cdot \begin{pmatrix}24 \\ -2 \\ 11\end{pmatrix} \\
+    \text{$g$ in $E$ eingesetzt} \\
+    -(2 + 24r) - (-2r) + 2(2 + 11r) + 7 = 0 \quad L = \{\} \quad\Rightarrow\quad g \parallel E \\\\
+    g' \colon \vv{x} = \begin{pmatrix}2 \\ 0 \\ 2\end{pmatrix} + r \cdot \begin{pmatrix}-1 \\ -1 \\ 2\end{pmatrix} \\
+    \text{$g'$ in $E$ eingesetzt} \\
+    \begin{pmatrix}1 - r \\ -6 - r \\ 2 + 2r\end{pmatrix} \cdot \begin{pmatrix}-1 \\ -1 \\ 2\end{pmatrix} = 0 \quad\Rightarrow\quad r = -\frac{3}{2} \qquad S(3.5|1.5|-1) \\
+    d(g, E) = d(D, E) = |\vv{DS}| = \sqrt{1.5^2 + 1.5^2 + (-3)^2} = \frac{9}{\sqrt{6}}
+  \end{gather*}
+  \item [c]
+  \begin{gather*}
+    F \colon \vv{x} = \begin{pmatrix}2 \\ 0 \\ 2\end{pmatrix} + r \cdot \begin{pmatrix}24 \\ -2 \\ 11\end{pmatrix} + s \cdot \begin{pmatrix}-1 \\ -1 \\ 2\end{pmatrix} \\
+    h \colon \vv{x} = \begin{pmatrix}3.5 \\ 1.5 \\ -1\end{pmatrix} + r \cdot \begin{pmatrix}24 \\ -2 \\ 11\end{pmatrix}
+  \end{gather*}
+  \item [d]
+  \begin{gather*}
+    k \colon \vv{x} = \begin{pmatrix}1 \\ 6 \\ 0\end{pmatrix} + r \cdot \begin{pmatrix}3 \\ 1 \\ 2\end{pmatrix} \\
+    E_k \colon \left(\begin{pmatrix}x_1 \\ x_2 \\ x_3\end{pmatrix} - \begin{pmatrix}2 \\ 7 \\ 1\end{pmatrix}\right) \cdot \begin{pmatrix}3 \\ 1 \\ 2\end{pmatrix} \qquad E_k \colon 3x_1 + x_2 + 2x_3 - 15 = 0 \\
+    \text{$g$ in $E_k$ eingesetzt} \\
+    3(1 + 3r) + (6 + r) + 2(2r) - 15 = 0 \quad\Rightarrow\quad r = \frac{3}{7} \\
+    \vv{OF} = \begin{pmatrix}1 \\ 6 \\ 0\end{pmatrix} + \frac{3}{7} \cdot \begin{pmatrix}3 \\ 1 \\ 2\end{pmatrix} = \begin{pmatrix}\frac{16}{7} \\[0.4em] \frac{45}{7} \\[0.4em] \frac{6}{7}\end{pmatrix} \\
+    |\vv{CF}| = \sqrt{(\frac{16}{7} - 2)^2 + (\frac{45}{7} - 7)^2 + (\frac{6}{7} - 1)^2} \approx 0.65465 \\\\
+    \triangle ABC = \frac{1}{2} \cdot |\vv{AB}| \cdot |\vv{CF}| \approx \frac{1}{2} \cdot 3.74166 \cdot 0.65465 \approx 1.22474
+  \end{gather*}
+  \item [e]
+  \begin{gather*}
+    V = \frac{1}{3} \cdot \triangle ABC \cdot d(D, E) \approx \frac{1}{3} \cdot 1.22474 \cdot \frac{9}{\sqrt{6}} \approx 1.5
+  \end{gather*}
+\end{exercise}
+\begin{exercise}{382/15}
+  \begin{gather*}
+    \vv{a} = \begin{pmatrix}2 \\ 1 \\ 0\end{pmatrix} \qquad \vv{b} = \begin{pmatrix}2 \\ -4 \\ 0\end{pmatrix} \qquad \vv{c} = \begin{pmatrix}0 \\ 0 \\ 3\end{pmatrix}
+  \end{gather*}
+  \item [a]
+  \begin{gather*}
+    \vv{a} \cdot \vv{b} = 2 \cdot 2 + 1 \cdot (-4) + 0 \cdot 0 = 0 \\
+    \vv{b} \cdot \vv{c} = 2 \cdot 0 + (-4) \cdot 0 + 0 \cdot 3 = 0 \\
+    \vv{c} \cdot \vv{a} = 0 \cdot 2 + 0 \cdot 1 + 3 \cdot 0 = 0
+  \end{gather*}
+  \item [b]
+  \begin{gather*}
+    \vv{OM} = \frac{1}{2}\vv{a} + \frac{1}{2}\vv{b} + \frac{1}{2}\vv{c} = \begin{pmatrix}2 \\ -1 \\ 1.5\end{pmatrix} \\
+    r = |\vv{OM}| = \sqrt{2^2 + (-1.5)^2 + 1.5^2} = \sqrt{8.5} \approx 2.91548 \\\\
+    K_b \colon \left(\vv{x} - \begin{pmatrix}2 \\ -1.5 \\ 1.5\end{pmatrix}\right)^2 = 8.5
+  \end{gather*}
+  \item [c]
+  \begin{gather*}
+    K \colon \left(\vv{x} - \begin{pmatrix}2 \\ -1.5 \\ 1.5\end{pmatrix}\right)^2 = 25 \\\\
+    \vv{OM_{ab}} = \vv{OM} \pm \frac{1}{2}\vv{c} = \begin{pmatrix}2 \\ -1.5 \\ 0\end{pmatrix} \text{ bzw. } \begin{pmatrix}2 \\ -1.5 \\ 3\end{pmatrix} \\
+    r_{ab} = \sqrt{r^2 - \left|\frac{1}{2}\vv{c}\right|^2} = \sqrt{22.75} \approx 4.76970 \\\\
+    \vv{OM_{bc}} = \vv{OM} \pm \frac{1}{2}\vv{a} = \begin{pmatrix}1 \\ -2 \\ 1.5\end{pmatrix} \text{ bzw. } \begin{pmatrix}3 \\ -1 \\ 1.5\end{pmatrix} \\
+    r_{bc} = \sqrt{r^2 - \left|\frac{1}{2}\vv{a}\right|^2} = \sqrt{23.75} \approx 4.87340 \\\\
+    \vv{OM_{ca}} = \vv{OM} \pm \frac{1}{2}\vv{b} = \begin{pmatrix}1 \\ 0.5 \\ 1.5\end{pmatrix} \text{ bzw. } \begin{pmatrix}3 \\ -3.5 \\ 1.5\end{pmatrix} \\
+    r_{ca} = \sqrt{r^2 - \left|\frac{1}{2}\vv{b}\right|^2} = \sqrt{20} \approx 4.47214
+  \end{gather*}
+\end{exercise}