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

Rozważmy wektor A, którego składowe x i y są reprezentowane w postaci wektorów jednostkowych, i i j. Tutaj wektory jednostkowe mają bezwymiarową wielkość jeden.
Ponieważ wielkość dowolnego składnika wektora jest zawsze wielkością dodatnią, reprezentowaną przez skalary, A można wyrazić jako wektor kartezjański.
W tym przypadku używany jest prawoskrętny, prostokątny układ współrzędnych. Kciuk prawej ręki jest skierowany w stronę dodatniej osi z, a palce zwijają się od dodatniej osi x w kierunku dodatniej osi y.
Wektor 3-wymiarowy można przedstawić w postaci prostokątnych współrzędnych kartezjańskich za pomocą wektorów jednostkowych i, j i k. Kierunek tych wektorów jest reprezentowany w zależności od osi dodatniej lub ujemnej.
Wektor jest reprezentowany jako suma wektorowa jego poszczególnych składników, a jego wielkość jest wyrażana jako dodatni pierwiastek kwadratowy z sumy kwadratów jego składników.
Operacje algebry wektorowej są uproszczone przez przedstawienie wektora w postaci kartezjańskiej. Oddziela jego wielkość i kierunek wzdłuż osi za pomocą notacji wektora jednostkowego.

<ol>
	<li>Hibbeler, R.C. (2016). Mechanika inżynierska \u2012 Statyka i dynamika. Hoboken, New Jersey: Pearson Prentice Hall. s. Strona nr- 34, 44, 45.</li>
</ol>

notacja wektorowa kartezjańska

Notacja wektorowa kartezjańska stanowi cenne narzędzie w inżynierii mechanicznej do reprezentowania wektorów w trójwymiarowej przestrzeni, umożliwiając wykonywanie operacji na wektorach, takich jak wyznaczanie gradientu, dywergencji i wirowania, a także wyrażanie wielkości fizycznych, takich jak przemieszczenie, prędkość, przyspieszenie i siła. Dzięki użyciu notacji wektorowej kartezjańskiej, inżynierowie mogą łatwiej analizować i rozwiązywać problemy w różnych dziedzinach inżynierii mechanicznej, w tym w dynamice, kinematyce i mechanice płynów. Notacja ta reprezentuje wektor za pomocą trzech składowych wzdłuż osi x, y i z odpowiednio.

Na przykład, załóżmy, że mamy wektor A skierowany w kierunku (3, -4, 5). W takim przypadku można go przedstawić za pomocą notacji wektorowej kartezjańskiej jako A = 3i - 4j + 5k, gdzie i, j i k są wektorami jednostkowymi wzdłuż osi x, y i z. Wektory jednostkowe są zdefiniowane jako i = (1, 0, 0), j = (0, 1, 0) i k = (0, 0, 1).

Notacja wektorowa kartezjańska może być używana do wykonywania różnych operacji na wektorach, takich jak dodawanie, odejmowanie i mnożenie przez skalar. Na przykład, jeśli mamy dwa wektory, A = 3i - 4j + 5k i B = 2i + 7j - 3k, możemy je dodać za pomocą notacji wektorowej kartezjańskiej w następujący sposób:

<img alt="Equation 1" src="/files/ftp_upload/14232/14232_ Equation_1v2.svg" style="text-align: center; height: 13px;" />

Również możemy je odjąć jak poniżej:

<img alt="Equation 2" src="/files/ftp_upload/14232/14232_ Equation_2v2.svg" style="text-align: center; height: 13px;" />

Notacja wektorowa kartezjańska

Kartezjańska notacja wektorowa jest cennym narzędziem w inżynierii mechanicznej do reprezentowania wektorów w przestrzeni trójwymiarowej, wykonywania operacji wektorowych ...

Education

JoVE Core

Mechanical Engineering

Inżynieria mechaniczna

Unit 1

Jednostka 1

Force Vectors

Wektory siły

KEY TERMS AND CONCEPTS

Kluczowe terminy i pojęcia

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

scalar and vectors

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force classification

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two-dimensional force system: problem solving

two-dimensional-force-system-problem-solving

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scalar notation

Vector Components for Force Calculations

Scalar notation helps simplify vector calculations by breaking a vector into separate x and y parts. When vectors are added or subtracted, each component can be handled on its own using scalar notation. This makes force problems easier to solve.

cartesian vector notation

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,...

direction cosines of a vector

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Direction cosines describe how a vector is oriented relative to the x, y, and z axes. For a vector A written in Cartesian form with i, j, and k unit vectors, this idea helps connect the vector’s components to its direction in space. The magnitude of vector A is the square root of the sum of the squares of its components. Its direction is described by the coordinate direction angles α, β, and γ with the x, y, and z axes. These angles can be found by projecting vector A onto each axis. The...

three-dimensional force system

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A three-dimensional force system describes forces acting in x, y, and z directions. In mechanical engineering, this framework helps students and engineers study how objects and structures behave in three-dimensional space. One common example is a force acting on a particle in space. That force can be split into three parts: the x-component, the y-component, and the z-component. Vector mathematics is used to calculate each component, and together they make up the full three-dimensional force...

three-dimensional force system:problem solving

three-dimensional-force-systemproblem-solving

Resolving Forces in 3D Coordinates

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position vectors

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dot product

Using Dot Product to Find Work and Angle

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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