Name: adding and multiplying vectors
Uploaded: 2023-09-22T11:49:19.000Z
Duration: 1 min 20 s
Channel: JoVE

Betrachten Sie zwei Kraftvektoren in einem zweidimensionalen Koordinatensystem.
Diese Vektoren können mit dem Parallelogrammgesetz addiert werden, bei dem die Schwänze der nebenläufigen Vektoren miteinander verbunden und Linien parallel zu jedem Vektor gezeichnet werden, um ein Parallelogramm zu bilden.
Der resultierende Vektor kann erhalten werden, indem eine Diagonale von den Schwänzen bis zum Schnittpunkt der Linien gezeichnet wird.
Auf ähnliche Weise können die Vektoren mit der Dreiecksregel addiert werden, wobei zwei Vektoren als zwei Seiten des Dreiecks angeordnet werden, die von Kopf bis Ende verbunden sind, und die dritte Seite des Dreiecks die Größe und Richtung des resultierenden Vektors darstellt.
Die Vektorsubtraktion ähnelt dem Addieren der Umkehrung eines Vektors zu einem anderen. Als Ergebnis wird die Differenz zwischen zwei Vektoren nach den gleichen Regeln der Vektoraddition erhalten.
Wenn ein Vektor mit einem Skalar multipliziert wird, variiert die Größe des Vektors um den Betrag der skalaren Größe.
Wenn der Skalar positiv ist, bleibt die Richtung des Vektors gleich, und wenn er negativ ist, wird die Richtung umgekehrt.

<ol>
	<li>Hibbeler, R.C. (2016). Technische Mechanik \u2012 Statik und Dynamik. Hoboken, New Jersey: Pearson Prentice Hall. Seiten 18 -19</li>
</ol>

vektoroperationen

Vektoren sind physikalische Größen, die sowohl Betrag als auch Richtung haben. Die Vektoroperationen umfassen Addition, Subtraktion und skalare Multiplikation. Ein Vektor, der mit einem skalaren Wert multipliziert wird, wird als skalare Multiplikation bezeichnet. Das Ergebnis ist ein neuer Vektor mit einem anderen Betrag. Wenn der Skalar positiv ist, bleibt die Richtung des Vektors gleich, aber wenn er negativ ist, wird die Richtung des Vektors umgekehrt. Zum Beispiel ergibt das Produkt aus Masse und Geschwindigkeit den Impuls. Hier wird die skalare Größe Masse mit der vektoriellen Größe Geschwindigkeit multipliziert, was zur vektoriellen Größe Impuls führt. Die Richtung des resultierenden Impulsvektors ist in dieselbe Richtung wie der Geschwindigkeitsvektor. Die beiden Vektoren können geometrisch mit dem Parallelogramm-Gesetz der Vektoraddition addiert werden. Bei Verwendung dieses Gesetzes werden die beiden Vektoren als benachbarte Seiten eines Parallelogramms dargestellt. Die Anfangspunkte beider Vektoren werden am Ursprung platziert und parallele Linien zu jedem Vektor werden gezeichnet, um ein Parallelogramm zu bilden. Die Diagonale des Parallelogramms wird dann vom Ursprung zur gegenüberliegenden Ecke des Parallelogramms gezeichnet, um den resultierenden Vektor darzustellen. Der Betrag und die Richtung des resultierenden Vektors können durch Messen der Länge und der Winkel der Diagonale des Parallelogramms bestimmt werden. Das Parallelogramm-Gesetz kann in einem speziellen Szenario vereinfacht werden, in dem die beiden Vektoren kollinear sind, d.h. sie haben die gleiche Wirkungslinie. In diesem Fall kann der resultierende Vektor durch eine algebraische oder skalare Addition der beiden Vektoren erhalten werden. Eine andere Methode zur grafischen Addition von zwei Vektoren ist das Dreiecksgesetz der Vektoraddition. Das Dreiecksgesetz der Vektoraddition besagt, dass wenn zwei Vektoren kopf-an-Schwanz als zwei Seiten eines Dreiecks angeordnet sind, die dritte Seite des Dreiecks den Betrag und die Richtung des resultierenden Vektors darstellt. Die Vektoraddition unter Verwendung dieses Gesetzes beinhaltet das Platzieren des ersten Vektors am Ursprung und das Platzieren des zweiten Vektors kopf-an-Schwanz mit dem ersten Vektor. Die dritte Seite des Dreiecks vom Ursprung bis zum Ende des zweiten Vektors stellt den resultierenden Vektor dar. Der Vorgang der Vektorsubtraktion beinhaltet das Hinzufügen der Umkehr (negativ) eines Vektors zu einem anderen Vektor. Das bedeutet, dass die Regeln der Vektoraddition angewendet werden können, um die Differenz zwischen den beiden Vektoren zu erhalten.

Vektoroperationen

Vektoren sind physikalische Größen, die sowohl Größe als auch Richtung haben. Zu den Vektoroperationen gehören Addition, Subtraktion und skalare Multiplikation.
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JoVE Core

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Maschinenbau

Unit 1

Einheit 1

Force Vectors

Kraftvektoren

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

Vector Operations

adding and multiplying vectors

Vectors are quantities with both magnitude and direction, and their operations include addition, subtraction, and scalar multiplication. Scalar multiplication means a vector is multiplied by a scalar value, which creates a new vector with a different magnitude. If the scalar is positive, the direction stays the same. If the scalar is negative, the direction reverses.

A common example is momentum, which comes from multiplying mass by velocity. In this case, mass is the scalar quantity and velocity is the vector quantity. The result is momentum, and its direction matches the direction of the velocity vector.

Two vectors can be added geometrically with the parallelogram law. In this method, the two vectors are drawn as adjacent sides of a parallelogram with both tails at the origin. Lines are then drawn parallel to each vector to complete the shape.

The diagonal of the parallelogram, drawn from the origin to the opposite corner, shows the resultant vector. Its magnitude and direction can be found by measuring the diagonal's length and angles. When the two vectors are collinear, meaning they share the same line of action, the resultant can be found by algebraic or scalar addition.

Another way to add vectors is the triangle law of vector addition. Here, the vectors are arranged head to tail as two sides of a triangle. The third side of the triangle, from the origin to the end of the second vector, represents the resultant vector.

Vector subtraction uses the reverse, or negative, of one vector added to another vector. This lets the rules of vector addition be used to find the difference between two vectors.

Adding and Multiplying Vectors

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vector operations

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key terms and concepts

Learn how vector addition, subtraction, and scalar multiplication work using parallelogram and triangle laws, plus a momentum example for high school science.

Adding and Multiplying Vectors | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

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Adding and Multiplying Vectors

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