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

Si consideri un vettore A con le sue componenti x e y rappresentate in termini di vettori unitari, i e j. Qui i vettori unitari hanno grandezza adimensionale di uno.
Poiché la grandezza di ogni componente vettoriale è sempre una quantità positiva, rappresentata da scalari, A può essere espresso come un vettore cartesiano.
In questo caso, viene utilizzato un sistema di coordinate rettangolari destrorso. Il pollice destro punta verso l'asse z positivo e le dita si arricciano dall'asse x positivo verso l'asse y positivo.
Un vettore tridimensionale può essere rappresentato in coordinate cartesiane rettangolari utilizzando vettori unitari i, j e k. La direzione di questi vettori è rappresentata a seconda degli assi positivo o negativo.
Un vettore è rappresentato come la somma vettoriale dei suoi singoli componenti e la sua grandezza è espressa come la radice quadrata positiva della somma dei quadrati dei suoi componenti.
Le operazioni di algebra vettoriale sono semplificate rappresentando il vettore nella forma cartesiana. Separa la sua grandezza e direzione lungo gli assi utilizzando la notazione vettoriale unitaria.

<ol>
	<li>Hibbeler, R.C. (2016). Ingegneria Meccanica \u2012 Statica e Dinamica. Hoboken, New Jersey: Pearson Prentice Hall. pp. Pagina n. 34, 44, 45.</li>
</ol>

notazione vettoriale di cartesio

La notazione vettoriale di Cartesio è uno strumento prezioso nell'ingegneria meccanica per rappresentare vettori nello spazio tridimensionale, eseguire operazioni vettoriali come determinare il gradiente, la divergenza e il rotore, ed esprimere grandezze fisiche come lo spostamento, la velocità, l'accelerazione e la forza. Utilizzando la notazione vettoriale di Cartesio, gli ingegneri possono analizzare e risolvere più facilmente problemi in vari settori dell'ingegneria meccanica, tra cui la dinamica, la cinematica e la meccanica dei fluidi. Questa notazione rappresenta un vettore in termini di tre componenti rispettivamente lungo gli assi x, y e z.

Ad esempio, supponiamo di avere un vettore A che punta nella direzione (3, -4, 5). In tal caso, può essere rappresentato utilizzando la notazione vettoriale di Cartesio come A = 3i - 4j + 5k, dove i, j e k sono vettori unitari rispettivamente lungo gli assi x, y e z. I vettori unitari sono definiti come i = (1, 0, 0), j = (0, 1, 0) e k = (0, 0, 1).

La notazione vettoriale di Cartesio può essere utilizzata per eseguire varie operazioni vettoriali, come l'addizione, la sottrazione e la moltiplicazione scalare. Ad esempio, se abbiamo due vettori, A = 3i - 4j + 5k e B = 2i + 7j - 3k, possiamo sommarli utilizzando la notazione vettoriale di Cartesio come segue:

<img alt="Equation 1" src="/files/ftp_upload/14232/14232_ Equation_1v2.svg" style="text-align: center; height: 13px;" />

Possiamo anche sottrarli come segue:

<img alt="Equation 2" src="/files/ftp_upload/14232/14232_ Equation_2v2.svg" style="text-align: center; height: 13px;" />

Notazione vettoriale di Cartesio

La notazione vettoriale cartesiana è uno strumento prezioso in ingegneria meccanica per rappresentare vettori nello spazio tridimensionale, eseguire operazioni vettoriali ...

Education

Didattica

JoVE Core

Mechanical Engineering

Ingegneria Meccanica

Unit 1

Unità 1

Force Vectors

Vettori di forza

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

Cartesian Vector Notation

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.

Vector Components in 3D Space

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scalar notation

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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,...

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three-dimensional force system:problem solving

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dot product

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The dot product connects vectors, work, and direction in mathematics and physics. In engineering, it is found by multiplying the magnitudes of two vectors and the cosine of the angle between them. It is shown with a dot symbol between the vectors. A real-world example is a vehicle pulling an object along the ground with a rope. If the rope is angled above the horizontal axis, the work done can be calculated with the dot product of the applied force and the object’s displacement. This makes the...

dot product: problem solving

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cross product

Cross Product Direction and Magnitude

The cross product is a vector operation in vector algebra that creates a third vector from two input vectors. Unlike the scalar product, it does not give a single number. The result is a vector perpendicular to both original vectors. The size, or magnitude, of the cross product depends on the magnitudes of both vectors and the sine of the angle between them. A larger angle between the vectors gives a larger cross product magnitude. The direction is found with the right-hand rule. If the...

key terms and concepts

Learn how Cartesian vector notation uses i, j, and k to show 3D vectors, describe force and motion, and perform addition, subtraction, and scalar multiplication.

Vector Components in 3D Space | 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.

mechanical engineering

Vector Components in 3D Space

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