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Zuletzt geändert von Anna Kukin am 2026/05/25 18:19
Von Version 41.1
bearbeitet von Holger Engels
am 2026/05/05 14:20
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Auf Version 10.2
bearbeitet von Dirk Tebbe
am 2026/04/27 16:19
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Zusammenfassung

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Titel
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1 -BPE 16.5 Gegenseitige Lage von Ebenen und Geraden
1 +BPE 16.5 Gegenseitige Lage von Ebenen
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Inhalt
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5 5  [[Kompetenzen.K5]] Ich kann eine Gleichung der Schnittgerade zwischen zwei Ebenen bestimmmen.
6 6  [[Kompetenzen.K5]] [[Kompetenzen.K4]] Ich kann Geraden und Ebenen angeben, die gegebene Lagebeziehungen erfüllen.
7 7  
8 -{{aufgabe id="Spiegeln" afb="I" kompetenzen="K5" quelle="Holger Engels" zeit="6"}}
9 -Gegeben ist die Ebene //E// mit der Gleichung {{formula}}E: 2x_1-x_2+x_3=4{{/formula}} und der Punkt {{formula}}P(0|0|0){{/formula}}. Bestimme den Punkt //P'//, der aus //P// durch Spiegelung an //E// entsteht.
8 +{{aufgabe id="Richtungsvektor und Spannvektoren" afb="I" kompetenzen="K1, K6" quelle="Holger Engels" zeit="3"}}
9 +Der Richtungsvektor einer Geraden //g// ist als Linearkombination der Spannvektoren der Ebene //E// darstellbar. Erläutere, was sich daraus über die Lage von //g// in Bezug auf //E// sagen lässt.
10 10  {{/aufgabe}}
11 11  
12 -{{aufgabe id="Aussagen" afb="II" kompetenzen="K1, K6" quelle="Holger Engels" zeit="6"}}
13 -(%class=abc%)
14 -1. Der Richtungsvektor einer Geraden //g// ist als Linearkombination der Spannvektoren der Ebene //E// darstellbar. Erläutere, was sich daraus über die Lage von //g// in Bezug auf //E// sagen lässt.
15 -1. Die Ebenen //E// und //F// teilen sich einen Spannvektor. Erläutere, was sich daraus über die Lage der beiden Ebenen zueinander sagen lässt.
16 -1. Die Ebenen //E// und //F// teilen sich den Punkt //P//. Erläutere, was sich daraus über die Lage der beiden Ebenen zueinander sagen lässt.
12 +{{aufgabe id="Gemeinsamer Spannvektor" afb="I" kompetenzen="K1, K6" quelle="Holger Engels" zeit="3"}}
13 +Die Ebenen //E// und //F// teilen sich einen Spannvektor. Erläutere, was sich daraus über die Lage der beiden Ebenen zueinander sagen lässt.
17 17  {{/aufgabe}}
18 18  
19 19  {{aufgabe id="Schnittgerade" afb="II" kompetenzen="K1, K6" quelle="Martin Stern, Dirk Tebbe" zeit="15"}}
20 -Es sind zwei Ebenen //E// und //F// gegeben durch {{formula}}E: 2x_1-3x_2+x_3=0{{/formula}} und {{formula}}F: 3x_1+2x_2=-1{{/formula}}.
17 +Es sind zwei Ebenen //E// und //F// gegeben:
18 +{{formula}}E: 2x_1-3x_2+x_3=0{{/formula}}
19 +{{formula}}F: 3x_1+2x_2=-1{{/formula}}
21 21  (%class=abc%)
21 +1. Zeige //E// und //F// schneiden sich.
22 22  1. Bestimme die Schnittgerade //g//.
23 -1. Welche besondere Lage haben die beiden Ebenen zueinander?
23 +1. Nimm Stellung zu folgender Aussage:
24 +//Die Projektion von E auf F ergibt g. Die Projektion von F auf E ergibt ebenfalls g//.
24 24  {{/aufgabe}}
25 25  
26 -{{aufgabe id="Lösungsmenge geometrisch" afb="II" kompetenzen="K6" quelle="Frauke Beckstette" zeit="12"}}
27 -Ordne den folgenden linearen Gleichungssystemen jeweils die passende Abbildung zu. Begründe deine Entscheidung.
28 -Visualisiere das verbliebene LGS analog.
29 29  
30 -(% class="abc horiz" %)
31 -1. (% style="vertical-align: top" %){{formula}}\begin{aligned}
32 - x_1 + x_2 &= 1 \\
33 - - 3x_2 &= 8 \\
34 - -x_1 + 2x_2 + x_3 &= 4
35 -\end{aligned}{{/formula}}
36 -1. (% style="vertical-align: top" %){{formula}}\begin{aligned}
37 - 3x_1 - 2x_2 + x_3 &= 7 \\
38 - -6x_1 + 4x_2 - 2x_3 &= 3 \\
39 - 15x_1 - 10x_2 + 5x_3 &= 5
40 -\end{aligned}{{/formula}}
41 -1. (% style="vertical-align: top" %){{formula}}\begin{aligned}
42 - 2x_1 - 2x_2 + 2x_3 &= 2 \\
43 - -2x_1 - 6x_2 + 2x_3 &= 0 \\
44 - 2x_1 + 2x_2 &= 1
45 -\end{aligned}{{/formula}}
46 -1. (% style="vertical-align: top" %){{formula}}\begin{aligned}
47 - x_1 + 3x_2 - 2x_3 &= 5 \\
48 - -2x_1 - 6x_2 + 4x_3 &= 1 \\
49 - 2x_1 + x_3 &= 3
50 -\end{aligned}{{/formula}}
51 -
52 -[[image:3 Ebenen A.svg||width="200"]][[image:3 Ebenen B.svg||width="200"]][[image:3 Ebenen C.svg||width="200"]]
53 -{{/aufgabe}}
54 -
55 -{{aufgabe id="Schnittpunkt Ebene Gerade" afb="I, II" kompetenzen="K2, K4, K5" quelle="[[IQB e.V.>>https://www.iqb.hu-berlin.de/media/exercise_files/Abituraufgaben_Mathematik/2017MerhoehtAAGLAA211_Aufgabe.pdf]]" zeit="15" niveau="e" tags="iqb" cc="BY"}}
56 -Gegeben sind die Ebene {{formula}}E: x_1 + x_2 + 2x_3 = 4{{/formula}} und \\
57 -die Gerade {{formula}}g: \vec{x} = \begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix} + \lambda \cdot \begin{pmatrix} 2 \\ -1 \\ -3 \end{pmatrix}{{/formula}} mit {{formula}}\lambda \in \mathbb{R}{{/formula}}.
58 -(%class=abc%)
59 -1. Zeichne die Schnittgerade von {{formula}}E{{/formula}} mit der {{formula}}x_2x_3{{/formula}}-Ebene in ein Koordinatensystem ein.
60 -1. Berechne die Koordinaten des Schnittpunktes von {{formula}}E{{/formula}} und {{formula}}g{{/formula}}.
61 -{{/aufgabe}}
3 Ebenen A.svg
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3 Ebenen B.svg
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3 Ebenen C.svg
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3 Ebenen X.svg
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