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

Considérons un vecteur A avec ses composantes x et y représentées en termes de vecteurs unitaires, i et j. Ici, les vecteurs unitaires ont une norme sans dimension de un.
Comme l’intensité de toute composante vectorielle est toujours une quantité positive, représentée par des scalaires, A peut être exprimé comme un vecteur cartésien.
Ici, un système de coordonnées rectangulaires à droite est utilisé. Le pouce de la main droite pointe vers l’axe z positif et les doigts s’enroulent de l’axe x positif vers l’axe y positif.
Un vecteur tridimensionnel peut être représenté en coordonnées cartésiennes rectangulaires à l’aide de vecteurs unitaires i, j et k. La direction de ces vecteurs est représentée en fonction des axes positifs ou négatifs.
Un vecteur est représenté par la somme vectorielle de ses composantes individuelles, et sa norme est exprimée par la racine carrée positive de la somme des carrés de ses composantes.
Les opérations d’algèbre vectorielle sont simplifiées en représentant le vecteur sous forme cartésienne. Il sépare sa norme et sa direction le long des axes à l’aide de la notation vectorielle unitaire.

<ol>
	<li>Hibbeler, R.C. (2016). Génie mécanique \u2012 Statique et dynamique. Hoboken, New Jersey : Pearson Prentice Hall. pp. Page n° 34, 44, 45.</li>
</ol>

notation vectorielle cartésienne

La notation vectorielle cartésienne est un outil précieux en génie mécanique pour représenter des vecteurs dans l'espace tridimensionnel, effectuer des opérations vectorielles telles que la détermination du gradient, la divergence et le tourbillon, et exprimer des grandeurs physiques telles que le déplacement, la vitesse, l'accélération et la force. En utilisant la notation vectorielle cartésienne, les ingénieurs peuvent plus facilement analyser et résoudre des problèmes dans différents domaines du génie mécanique, notamment la dynamique, la cinématique et la mécanique des fluides. Cette notation représente un vecteur en termes de trois composantes le long des axes x, y et z, respectivement. Par exemple, supposons que nous ayons un vecteur A qui pointe dans la direction (3, −4, 5). Dans ce cas, il peut être représenté à l'aide de la notation vectorielle cartésienne comme A = 3i - 4j + 5k, où i, j et k sont les vecteurs unitaires le long des axes x, y et z, respectivement. Les vecteurs unitaires sont définis comme i = (1, 0, 0), j = (0, 1, 0), et k = (0, 0, 1). La notation vectorielle cartésienne peut être utilisée pour effectuer différentes opérations vectorielles, telles que l'addition, la soustraction et la multiplication scalaire. Par exemple, si nous avons deux vecteurs, A = 3i - 4j + 5k et B = 2i + 7j - 3k, nous pouvons les additionner en utilisant la notation vectorielle cartésienne comme suit : <img alt="Equation 1" src="/files/ftp_upload/14232/14232_ Equation_1v2.svg" style="text-align: center; height: 13px;"> Nous pouvons également les soustraire comme suit : <img alt="Equation 2" src="/files/ftp_upload/14232/14232_ Equation_2v2.svg" style="text-align: center; height: 13px;">

Notation vectorielle cartésienne

La notation vectorielle cartésienne est un outil précieux en génie mécanique pour représenter des vecteurs dans l’espace tridimensionnel, effectuer des opérations vectorielles ...

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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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