Betrag Eines Vectors Berechnen Beispiel Essay

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Betrag eines Vektors

Unter dem Betrag eines Vektors versteht man in der Mathematik nichts anderes als die Länge eines Vektors.

Ist ein Vektor \(\vec{v}\) gegeben

\(\vec{v}= \begin{pmatrix} x \\ y \end{pmatrix}\)

dann berechnet sich der Betrag des Vektors zu

\(\left|\vec{v}\right| = \sqrt{x^2 + y^2} \)

Länge eines Vektors berechnen - Beispiel

Gegeben ist der Vektor \(\vec{v}\)

\(\vec{v} = \begin{pmatrix} 1 \\ 2 \\ 2 \end{pmatrix}\)

Die Länge des Vektors berechnet sich dann zu

\( \left|\vec{v}\right| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3;\)

Antwort: Der Vektor hat einen Betrag von 3 Einheiten.

Beispiel - Graphisch

Einige von euch interessieren sich sicherlich dafür, wie man auf die Formel für die Berechnung des Betrags eines Vektors kommt. Nun, das ist gar nicht so kompliziert.

Stellen wir uns folgende Situation vor: Wir haben einen Vektor \(\vec{a}\) gegeben (vgl. Abbildung)

\(\vec{a} = \begin{pmatrix} 3 \\ 3 \end{pmatrix} \)

und kennen die Formel zur Berechnung der Länge nicht. Jetzt teilen wir den Vektor in zwei Vektoren auf, deren Länge wir kennen

\(\vec{x_a} = \begin{pmatrix} 3 \\ 0 \end{pmatrix} \qquad \vec{y_a} = \begin{pmatrix} 0 \\ 3 \end{pmatrix}\)

(Wer es nicht sofort sieht: \(\vec{x_a}\) und \(\vec{y_a}\) haben jeweils die Länge 3.)

Addiert man diese beiden Vektoren, erhalten wir wieder unseren Vektor \(\vec{a}\)

\(\vec{x_a} + \vec{y_a} = \begin{pmatrix} 3 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 3 \end{pmatrix} = \begin{pmatrix} 3 \\ 3 \end{pmatrix} = \vec{a}\)

"Toll, was hat uns das jetzt gebracht?", werden sich einige von euch fragen. Wie wir aus der Vektoraddition wissen, bilden diese drei Vektoren ein Dreieck. Zwei Seitenlängen kennen wir und die dritte ist gesucht...!? Das schreit doch förmlich nach dem berühmten Satz des Pythagoras:

\(a^2 + b^2 = c^2\)

Mit Hilfe des Satzes des Pythagoras kannst du die dritte Seite eines Dreiecks berechnen, wenn du bereits zwei Seiten kennst. Dazu muss man nur die Formel nach einer Seite (z.B. c) auflösen

\(c = \sqrt{a^2 + b^2}\)

Kommt dir die Formel bekannt vor? Sieht der Formel zur Berechnung des Betrags eines Vektors doch ziemlich ähnlich, oder?

\(\left|\vec{v}\right| = \sqrt{x^2 + y^2}\)

Für unser Beispiel gilt

\(\left|\vec{a}\right| = \sqrt{3^2 + 3^2} = \sqrt{9+9} = \sqrt{18} \approx 4,24\)

Diese Kapitel diente als eine Art Herleitung für die Formel. Wenn du den Zusammenhang zwischen dem Satz des Pythagoras und der Formel zur Berechnung der Länge eines Vektors nicht sofort verstanden hast, ist das nicht schlimm. Dein Lehrer in der Schule will nur, dass du den Betrag eines Vektors berechnen kannst. Aus diesem Grund solltest du dir die folgenden Aufgaben sehr genau angucken.

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