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

位置矢量是一种固定矢量，用于确定空间中某一点相对于另一点的位置。
假设一个螺环固定在墙上，一根缆绳 OP 与该螺环相连。建立一个笛卡尔坐标系，以螺环为坐标原点，沿各坐标轴设置分量矢量，以确定缆绳上点 P 的位置。
从原点指向点 P 的位置矢量 r 可以用笛卡尔矢量形式表示。
为了直观表示笛卡尔向量，可以考虑固定在墙上的一个支架，并将一根绳索连接到该支架上。
设 nA 和 nB 为从原点指向点 A 和点 B 的位置矢量。
因此，从点 A 到点 B 的合位置矢量 n 可通过三角形法则，并将 nA 和 nB 表示为笛卡尔矢量形式来求得。
i、j 和 k 是沿位置矢量 n 分量方向的单位矢量，该位置矢量 n 由 B 的坐标减去 A 的坐标得到。

<ol>
	<li>罗素 C. 希伯勒。（2016）. 工程力学:静力学与动力学。皮尔森。（第 56,57 页）</li>
</ol>

位置向量

位置向量是数学中的基本概念，能够用来确定空间中一个点相对于另一个点的位置。它是用来描述两个点之间的方向向量和距离向量。位置向量在数学和科学领域是非常有用的，因为它们有助于表示空间关系并对其进行简化计算。

例如，我们想要确定一个点P(x, y, z)相对于坐标原点O的位置。在这种情况下，我们可以定义一个位置向量r，它从原点O延伸到点P。我们可以用笛卡尔向量的形式来表示这个向量：r = xi + yj + zk，其中i，j和k分别是x，y和z方向上的单位向量。位置向量r给出了从O点到P点向量的方向和大小。

思考一个从A点到B点的位置向量在空间中的方向。这个向量可以用符号r来进行表示。我们还可以使用两个下标来表示向量的起始点和终点。因此，我们也可以把r表示为rAB。请注意，如果位置向量从坐标原点进行延伸，那么它们将会仅带有一个下标，如rA和rB。位置向量rAB则可以通过rA和rB之间的表达式rAB = rB - rA = (xB - xA)i + (yB - yA)j + (zB - zA)k来得出。

例如，要建立从点A到B的位置向量，将尾部坐标A(1m, 0m, -3m)从头部坐标B(-2m, 2m, 3m)中减去，则可以得到rAB={ -3i + 2j + 6k} m。

位置矢量是数学中的一个基本概念，用于确定空间中一个点相对于另一个点的位置。它是一个……

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JoVE Core

Mechanical Engineering

机械工程

Unit 1

第一单元

Force Vectors

力矢量

KEY TERMS AND CONCEPTS

关键术语与概念

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

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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

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

Using Position Vectors to Locate Points

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