Einheitsvektor

Ein Einheitsvektor ist in der analytischen Geometrie ein Vektor der Länge Eins. In der linearen Algebra und der Funktionalanalysis wird der Begriff der Länge auf allgemeine Vektorräume zum Begriff der Norm verallgemeinert. Ein Vektor in einem normierten Vektorraum, das heißt einem Vektorraum, auf dem eine Norm definiert ist, heißt Einheitsvektor oder normierter Vektor, wenn seine Norm Eins beträgt.

Definition

Ein Element eines normierten Vektorraumes heißt Einheitsvektor, wenn gilt. Einheitsvektoren werden oft mit einem Zirkumflex gekennzeichnet ().[1]

Einordnung

Einen gegebenen, vom Nullvektor verschiedenen Vektor kann man normieren, indem man ihn durch seine Norm (= seinen Betrag) dividiert:

Dieser Vektor ist der Einheitsvektor, der in dieselbe Richtung wie zeigt. Er spielt z. B. eine Rolle beim Gram-Schmidtschen Orthogonalisierungsverfahren oder der Berechnung der Hesseschen Normalform.

Die Elemente einer Basis (= Basisvektoren) werden oft als Einheitsvektoren gewählt, denn durch die Verwendung von Einheitsvektoren werden viele Rechnungen vereinfacht. Zum Beispiel ist in einem euklidischen Raum das Standardskalarprodukt zweier Einheitsvektoren gleich dem Kosinus des Winkels zwischen den beiden.

Endlichdimensionaler Fall

Kanonische Einheitsvektoren in der euklidischen Ebene

In den endlichdimensionalen reellen Vektorräumen besteht die am häufigsten bevorzugte Standardbasis aus den kanonischen Einheitsvektoren

.

Fasst man die kanonischen Einheitsvektoren zu einer Matrix zusammen, erhält man eine Einheitsmatrix.

Die Menge der kanonischen Einheitsvektoren des bildet bezüglich des kanonischen Skalarprodukts eine Orthonormalbasis, d. h. je zwei kanonische Einheitsvektoren stehen senkrecht aufeinander (=„ortho“), alle sind normiert (=„normal“) und sie bilden eine Basis.

Beispiel

Die drei kanonischen Einheitsvektoren des dreidimensionalen Vektorraums werden in den Naturwissenschaften auch mit bezeichnet:

Unendlichdimensionaler Fall

In unendlichdimensionalen unitären Vektorräumen (= VR mit Skalarprodukt) bildet die (unendliche) Menge der kanonischen Einheitsvektoren zwar noch ein Orthonormalsystem, aber nicht notwendig eine (Vektorraum-)Basis. In Hilberträumen gelingt es jedoch durch Zulassung unendlicher Summen, jeden Vektor des Raumes darzustellen, man spricht deshalb weiter von einer Orthonormalbasis.

Siehe auch

Wiktionary: Einheitsvektor – Bedeutungserklärungen, Wortherkunft, Synonyme, Übersetzungen

Einzelnachweise

  1. Principles Of Physics: A Calculus-based Text, Band 1, Raymond A. Serway, John W. Jewett, Verlag: Cengage Learning, 2006, ISBN 9780534491437, S. 19, eingeschränkte Vorschau in der Google-Buchsuche

Literatur

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