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Zuletzt geändert von Anna Kukin am 2026/05/25 18:19
Von Version 42.2
bearbeitet von Holger Engels
am 2026/05/07 21:02
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Auf Version 33.2
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Zusammenfassung

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Seiteneigenschaften
Titel
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1 -BPE 16.5 Gegenseitige Lage von Ebenen und Geraden
1 +BPE 16.5 Gegenseitige Lage von Ebenen
Inhalt
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1 1  {{seiteninhalt/}}
2 2  
3 3  [[Kompetenzen.K5]] [[Kompetenzen.K6]] Ich kann die gegenseitige Lage von Ebenen und Geraden untersuchen.
4 -[[Kompetenzen.K5]] Ich kann die Koordinaten des Schnittpunktes von Gerade und Ebene bestimmen.
5 -[[Kompetenzen.K5]] Ich kann eine Gleichung der Schnittgerade zwischen zwei Ebenen bestimmmn.
4 +[[Kompetenzen.K5]] Ich kann die Koordinaten des Schnittpunktes von Gerade und Ebene bestimmmen.
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.
10 -{{/aufgabe}}
11 -
12 12  {{aufgabe id="Aussagen" afb="II" kompetenzen="K1, K6" quelle="Holger Engels" zeit="6"}}
13 13  (%class=abc%)
14 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.
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16 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.
17 17  {{/aufgabe}}
18 18  
19 -{{aufgabe id="Aus Schnittgerade" afb="II" kompetenzen="K5" quelle="Holger Engels" zeit="4"}}
20 -Gib die Gleichungen zweier Ebenen an, die sich in der Geraden //g// mit {{formula}}g: \vec{x} = \begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix} + t \cdot \begin{pmatrix} 2 \\ -1 \\ -3 \end{pmatrix}{{/formula}} schneiden.
21 -{{/aufgabe}}
22 -
23 23  {{aufgabe id="Schnittgerade" afb="II" kompetenzen="K1, K6" quelle="Martin Stern, Dirk Tebbe" zeit="15"}}
24 24  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}}.
25 25  (%class=abc%)
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53 53   2x_1 + x_3 &= 3
54 54  \end{aligned}{{/formula}}
55 55  
56 -[[image:3 Ebenen A.svg||width="200"]][[image:3 Ebenen B.svg||width="200"]][[image:3 Ebenen C.svg||width="200"]]
48 +[[image:3 Ebenen Y.svg||width="200"]][[image:3 Ebenen C.svg||width="200"]][[image:3 Ebenen D.svg||width="200"]]
57 57  {{/aufgabe}}
58 -
59 -{{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"}}
60 -Gegeben sind die Ebene {{formula}}E: x_1 + x_2 + 2x_3 = 4{{/formula}} und \\
61 -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}}.
62 -(%class=abc%)
63 -1. Zeichne die Schnittgerade von {{formula}}E{{/formula}} mit der {{formula}}x_2x_3{{/formula}}-Ebene in ein Koordinatensystem ein.
64 -1. Berechne die Koordinaten des Schnittpunktes von {{formula}}E{{/formula}} und {{formula}}g{{/formula}}.
65 -{{/aufgabe}}
3 Ebenen A.svg
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3 Ebenen B.svg
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3 Ebenen D.svg
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3 Ebenen Y.svg
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