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Untersuche die gegenseitige Lage der beiden Ebenen
E: und F:
Berechne ggf. den Abstand.
Ebenen sind identisch
Ebenen sind parallel, Abstand der parallelen Ebenen
Ebenen schneiden sich

Ask by Byrd Ortiz. in Germany
Jan 21,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

Die beiden Ebenen sind identisch.

Solution

Die beiden Ebenen und sind gegeben durch:
Schritt 1: Überprüfung der Parallelität
Zunächst vergleichen wir die Normalenvektoren der beiden Ebenen:
  • Normalenvektor von :
  • Normalenvektor von :
Es ist ersichtlich, dass . Da die Normalenvektoren proportional sind, sind die Ebenen entweder identisch oder parallel.
Schritt 2: Überprüfung auf Identität
Um festzustellen, ob die Ebenen identisch sind, prüfen wir, ob die Gleichung der einen Ebene ein Vielfaches der anderen ist:
Multiplizieren wir die Gleichung von mit :
Dies entspricht genau der Gleichung von . Daher sind die Ebenen identisch.
Fazit:
Die beiden Ebenen sind identisch.
Antwort:
Ebenen sind identisch

Answered by UpStudy AI and reviewed by a Professional Tutor

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Extra Insights

Die beiden Ebenen und lassen sich durch die Gleichungen umformen. Wenn man die Gleichung von durch 3 teilt, erhält man , was exakt der Gleichung von entspricht. Das bedeutet, die beiden Ebenen sind identisch, da sie dieselbe geometrische Fläche im Raum beschreiben.
Da die Ebenen identisch sind, beträgt der Abstand zwischen ihnen . Wenn zwei Ebenen identisch sind, überlagern sie sich vollständig, was zu keinem Abstand führt – sie sind im Raum wie zwei Blätter in einem Buch, die perfekt übereinanderliegen.

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