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

Considérons deux vecteurs de force dans un système de coordonnées bidimensionnel.
Ces vecteurs peuvent être ajoutés à l’aide de la loi du parallélogramme, où les queues des vecteurs concurrents sont jointes et des lignes parallèles à chaque vecteur sont tracées pour former un parallélogramme.
Le vecteur résultant peut être obtenu en traçant une diagonale à partir des queues jusqu’au point d’intersection des droites.
De même, les vecteurs peuvent être ajoutés à l’aide de la règle du triangle, où deux vecteurs sont disposés comme deux côtés du triangle joints tête à queue, et le troisième côté du triangle représente l’intensité et la direction du vecteur résultant.
La soustraction vectorielle est similaire à l’addition de l’inverse d’un vecteur à un autre. Par conséquent, la différence entre deux vecteurs est obtenue en utilisant les mêmes règles d’addition de vecteurs.
Lorsqu’un vecteur est multiplié par un scalaire, l’intensité du vecteur varie en fonction de la quantité scalaire.
Si le scalaire est positif, alors la direction du vecteur reste la même, et s’il est négatif, la direction est inversée.

<ol>
	<li>Hibbeler, R.C. (2016). Génie mécanique \u2012 Statique et dynamique. Hoboken, New Jersey : Pearson Prentice Hall. p. 18 et 19</li>
</ol>

opérations vectorielles

Les vecteurs sont des quantités physiques qui possèdent à la fois une magnitude et une direction. Les opérations vectorielles comprennent l'addition, la soustraction et la multiplication scalaire. Un vecteur multiplié par une valeur scalaire est appelé multiplication scalaire. Le résultat obtenu est un nouveau vecteur avec une magnitude différente. Si le scalaire est positif, la direction du vecteur reste la même mais, si elle est négative, la direction du vecteur est inversée. Par exemple, le produit de la masse et de la vitesse donne l'impulsion. Ici, la quantité scalaire de masse est multipliée par la quantité vectorielle vitesse, ce qui donne la quantité vectorielle de l'impulsion. La direction du vecteur impulsion résultante est dans la même direction que le vecteur vitesse. Les deux vecteurs peuvent être ajoutés géométriquement en utilisant la loi du parallélogramme de l'addition vectorielle. Lorsqu'ils sont ajoutés en utilisant cette loi, les deux vecteurs sont représentés comme des côtés adjacents d'un parallélogramme. Les extrémités des deux vecteurs sont placées à l'origine, et les lignes parallèles à chaque vecteur sont dessinées pour former un parallélogramme. La diagonale du parallélogramme est ensuite tirée de l'origine à l'angle opposé du parallélogramme pour représenter le vecteur résultant. La magnitude et la direction du vecteur résultant peuvent être déterminées en mesurant la longueur et les angles de la diagonale du parallélogramme. La loi du parallélogramme peut être simplifiée dans un scénario spécial où les deux vecteurs sont colinéaires, c'est-à-dire qu'ils ont la même ligne d'action. Dans ce cas, le vecteur résultant peut être obtenu par une addition algébrique ou scalaire des deux vecteurs. Une autre méthode pour ajouter graphiquement deux vecteurs est la loi du triangle de l'addition vectorielle. La règle du triangle de l'addition vectorielle stipule que si deux vecteurs sont placés tête-bêche comme deux côtés d'un triangle, le troisième côté du triangle représente la magnitude et la direction du vecteur résultant. L'addition vectorielle en utilisant cette loi implique de placer le premier vecteur à l'origine et de placer le deuxième vecteur tête à queue avec le premier vecteur. Le troisième côté du triangle de l'origine jusqu'à l'extrémité du deuxième vecteur représente le vecteur résultant. Le processus de soustraction de vecteurs implique d'ajouter le renversement (négatif) d'un vecteur à un autre vecteur. Cela signifie que les règles de l'addition vectorielle peuvent être appliquées pour obtenir la différence entre les deux vecteurs.

Opérations vectorielles

Les vecteurs sont des grandeurs physiques qui ont à la fois une norme et une direction. Les opérations vectorielles comprennent l’addition, la soustraction et la multiplication scalaire.
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Education

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Mechanical Engineering

Génie mécanique

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Force Vectors

Vecteurs de force

KEY TERMS AND CONCEPTS

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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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