Name: vector components in 3d space
Uploaded: 2023-09-22T11:49:19.000Z
Duration: 1 min 28 s
Channel: JoVE

Betrachten Sie einen Vektor A mit seinen x- und y-Komponenten, die in Form von Einheitsvektoren i und j dargestellt werden. Hier haben die Einheitsvektoren die dimensionslose Größe eins.
Da die Größe einer Vektorkomponente immer eine positive Größe ist, die durch Skalare dargestellt wird, kann A als kartesischer Vektor ausgedrückt werden.
Hier wird ein rechtshändiges, rechteckiges Koordinatensystem verwendet. Der Daumen der rechten Hand zeigt auf die positive Z-Achse, und die Finger krümmen sich von der positiven X-Achse zur positiven Y-Achse.
Ein 3-dimensionaler Vektor kann in rechteckigen kartesischen Koordinaten mit Hilfe von i-, j- und k-Einheitsvektoren dargestellt werden. Die Richtung dieser Vektoren wird in Abhängigkeit von der positiven oder negativen Achse dargestellt.
Ein Vektor wird als Vektorsumme seiner einzelnen Komponenten dargestellt, und seine Größe wird als positive Quadratwurzel aus der Summe der Quadrate seiner Komponenten ausgedrückt.
Operationen der Vektoralgebra werden vereinfacht, indem der Vektor in der kartesischen Form dargestellt wird. Es trennt seine Größe und Richtung entlang der Achsen mit Hilfe der Einheitsvektornotation.

<ol>
	<li>Hibbeler, R.C. (2016). Technische Mechanik \u2012 Statik und Dynamik. Hoboken, New Jersey: Pearson Prentice Hall. S. Seite Nr. 34, 44, 45.</li>
</ol>

kartesische vektor−schreibweise

Kartesische Vektornotation ist ein wertvolles Werkzeug im Maschinenbau zur Darstellung von Vektoren im dreidimensionalen Raum, zur Durchführung von Vektoroperationen wie der Bestimmung des Gradienten, der Divergenz und der Rotation sowie zur Ausdrückung physikalischer Größen wie Verschiebung, Geschwindigkeit, Beschleunigung und Kraft. Durch die Verwendung der kartesischen Vektornotation können Ingenieure Probleme in verschiedenen Bereichen des Maschinenbaus, einschließlich Dynamik, Kinematik und Strömungsmechanik, leichter analysieren und lösen. Diese Notation stellt einen Vektor in Bezug auf drei Komponenten entlang der Achsen x, y und z dar.

Zum Beispiel, angenommen wir haben einen Vektor A, der in Richtung (3, −4, 5) zeigt. In diesem Fall kann er mit kartesischer Vektornotation als A = 3i − 4j + 5k dargestellt werden, wobei i, j und k Einheitsvektoren entlang der Achsen x, y und z sind. Die Einheitsvektoren sind definiert als i = (1, 0, 0), j = (0, 1, 0) und k = (0, 0, 1).

Kartesische Vektornotation kann verwendet werden, um verschiedene Vektoroperationen durchzuführen, wie Addition, Subtraktion und skalare Multiplikation. Zum Beispiel, wenn wir zwei Vektoren haben, A = 3i − 4j + 5k und B = 2i + 7j − 3k, können wir sie mit kartesischer Vektornotation wie folgt addieren:

<img alt="Equation 1" src="/files/ftp_upload/14232/14232_ Equation_1v2.svg" style="text-align: center; height: 13px;" />

Wir können sie auch wie folgt subtrahieren:

<img alt="Equation 2" src="/files/ftp_upload/14232/14232_ Equation_2v2.svg" style="text-align: center; height: 13px;" />

Kartesische Vektor−Schreibweise

Die kartesische Vektornotation ist ein wertvolles Werkzeug im Maschinenbau zur Darstellung von Vektoren im dreidimensionalen Raum, zur Durchführung von Vektoroperationen ...

Education

Lehre

JoVE Core

Mechanical Engineering

Maschinenbau

Unit 1

Einheit 1

Force Vectors

Kraftvektoren

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

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vector components in 3d space

Cartesian vector notation shows how to write vectors in three-dimensional space using x, y, and z components. It gives a clear way to represent direction and size for quantities used in mechanical engineering.

The notation uses the unit vectors i, j, and k for the x, y, and z axes. A vector pointing in the direction (3, −4, 5) can be written as A = 3i - 4j + 5k. The unit vectors are i = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1).

This form is useful for describing displacement, velocity, acceleration, and force. It also helps engineers work with dynamics, kinematics, and fluid mechanics. Cartesian vector notation can also support vector operations such as addition, subtraction, and scalar multiplication.

The same notation is also used with gradient, divergence, and curl. These operations are important in mechanical engineering calculations. For example, two vectors such as A = 3i - 4j + 5k and B = 2i + 7j - 3k can be added or subtracted using their components.

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Vector Components in 3D Space | JoVE K12

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Vector Components in 3D Space

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