Name: using position vectors to locate points
Uploaded: 2025-05-22T09:06:01.000Z
Duration: 1 min 29 s
Channel: JoVE

Um vetor de posição é um vetor fixo que localiza a posição de um ponto no espaço em relação a outro ponto.
Considere um olhal de parafuso fixado a uma parede e um cabo OP conectado a ele. Estabeleça um sistema de coordenadas cartesianas com olho de parafuso na origem e os vetores componentes ao longo dos eixos para localizar o ponto P no cabo.
Um vetor de posição r direcionado da origem ao ponto P pode ser expresso na forma vetorial cartesiana.
Para uma representação geral de um vetor cartesiano, considere um suporte fixado em uma parede e um cordão preso a ele.
Sejam nA e nB os vetores de posição que se estendem da origem aos pontos A e B.
Assim, o vetor de posição resultante n do ponto A a B pode ser obtido usando a regra do triângulo e expressando nA e nB na forma vetorial cartesiana.
i, j e k são os vetores unitários ao longo dos componentes do vetor de posição n formados pela subtração das coordenadas de A de B.

<ol>
	<li>Russell C. Hibbeler. (2016). Engenharia Mecânica: Estática e Dinâmica. Pearson. (Pág. Nº - 56,57)</li>
</ol>

vetores de posição

Um vetor de posição é um conceito fundamental em matemática que ajuda a determinar a posição de um ponto em relação a outro ponto no espaço. É um vetor que descreve a direção e a distância entre dois pontos. Vetores de posição são altamente úteis no campo da matemática e ciência, já que ajudam a representar relacionamentos espaciais e tornam os cálculos mais fáceis.

Por exemplo, queremos localizar um ponto P(x, y, z) em relação à origem das coordenadas O. Nesse caso, podemos definir um vetor de posição r, que se estende da origem O ao ponto P. Podemos expressar esse vetor na forma de vetor cartesiano como: r = xi + yj + zk, onde i, j, e k, são os vetores unitários nas direções x, y, e z, respectivamente. O vetor de posição r nos dá a direção e magnitude do vetor do ponto O ao ponto P.

Considere um vetor de posição direcionado do ponto A ao ponto B no espaço. Esse vetor pode ser representado pelo símbolo r. Também podemos nos referir a esse vetor com dois subscritos para indicar os pontos de onde e para onde ele é direcionado. Portanto, também podemos designar r como rAB. Por favor, note que se os vetores de posição se estendem da origem das coordenadas, eles são referidos apenas com um subscrito, como rA e rB. O vetor de posição rAB pode ser obtido a partir de rA e rB usando a expressão rAB = rB - rA= (xB - xA)i + (yB - yA)j + (zB - zA)k.

Por exemplo, para estabelecer um vetor de posição do ponto A ao B, as coordenadas da cauda A(1 m,  m0, -3 m) são subtraídas das coordenadas da cabeça B(-2 m, 2 m, 3 m), resultando em rAB={ -3i + 2j + 6k} m.

Vetores de Posição

Um vetor de posição é um conceito fundamental em matemática que ajuda a determinar a posição de um ponto em relação a outro ponto no espaço. É um ...

Education

JoVE Core

Mechanical Engineering

Engenharia Mecânica

Unit 1

Unidade 1

Force Vectors

Vetores de Força

KEY TERMS AND CONCEPTS

TERMOS-CHAVE E CONCEITOS

Position Vectors

using position vectors to locate points

Position vectors are used to locate points in space and show the direction and distance between two points. In math and science, they help describe spatial relationships and make calculations easier. A position vector can connect one point to another point or show a point’s location relative to the origin of coordinates.

For a point P(x, y, z) relative to the origin O, the position vector r runs from O to P. In Cartesian vector form, it is written as r = x i + y j + z k. Here, i, j, and k are unit vectors in the x, y, and z directions.

The vector r gives both the direction and the magnitude of the path from O to P. When a vector points from point A to point B, it may be written as rAB. Position vectors from the origin use one subscript only, such as rA and rB.

A vector from A to B can be found by subtracting the coordinates of A from the coordinates of B. The relation is rAB = rB - rA = (xB - xA)i + (yB - yA)j + (zB - zA)k. For example, if A is (1 m, 0 m, -3 m) and B is (-2 m, 2 m, 3 m), then rAB = {-3 i + 2 j + 6 k} m.

Using Position Vectors to Locate Points

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

Two-Dimensional Forces: Free-Body Analysis

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scalar notation

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cartesian vector notation

Vector Components in 3D Space

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three-dimensional force system

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dot product

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cross product

Cross Product Direction and Magnitude

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key terms and concepts

Learn how position vectors locate points in space, use i, j, k notation, and calculate a vector from one point to another with a clear example.

Using Position Vectors to Locate Points | JoVE K12

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mechanical engineering

Using Position Vectors to Locate Points

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