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

متجه الموضع هو متجه ثابت يحدد موضع نقطة في الفضاء بالنسبة لنقطة أخرى.
ضع في اعتبارك العين اللولبية المثبتة على الحائط وكابل OP متصل به. قم بإنشاء نظام إحداثيات ديكارتي مع العين اللولبية في الأصل والمتجهات المكونة على طول المحاور لتحديد موقع النقطة P على الكابل.
يمكن التعبير عن متجه الموضع r الموجه من الأصل إلى النقطة P في شكل المتجه الديكارتي.
للحصول على تمثيل عام للمتجه الديكارتي ، ضع في اعتبارك قوسا مثبتا على الحائط وسلك متصل به.
لنفترض أن nA و nB هما متجهان الموضع الممتدان من الأصل إلى النقطتين A و B.
إذن، يمكن الحصول على متجه الموضع الناتج n من النقطة A إلى B، باستخدام قاعدة المثلث والتعبير عن nA و nB في صورة المتجه الديكارتي.
i و j و k هي متجهات الوحدة على طول مكونات متجه الموضع n المكونة بطرح إحداثيات A من B.

<ol>
	<li>راسل سي هيبلر. (2016). الميكانيكا الهندسية: الإحصاء والديناميكيات. بيرسون. (الصفحة رقم - 56،57)</li>
</ol>

المتجهات الموضعية.

<div style="direction:rtl">

 
 المتجه الموضعي هو مفهوم أساسي في الرياضيات يساعد في تحديد موضع نقطة ما بالنسبة لنقطة أخرى في الفضاء. وهو متجه يصف
 الاتجاه والمسافة بين نقطتين. تُعد المتجهات الموضعية ذات أهمية كبيرة في مجالي الرياضيات والعلوم، حيث إنها تساعد في
 تمثيل العلاقات المكانية وتبسيط الحسابات. 

 
 على سبيل المثال، إذا أردنا تحديد موقع نقطةP(x,y,z) بالنسبة إلى نقطة أصل
 الإحداثيات O، يمكننا تعريف متجه الموضع r، الذي يمتد من الأصل O إلى النقطة P.
 يمكننا التعبير عن هذا المتجه في الشكل الاتجاهي الديكارتي على النحو التالي:
 r=xi+yj+zk
 حيث أن i، j، و k هي متجهات الوحدة في الاتجاهات x،
 y، و z على التوالي. يوفر متجه الموضع r كلاً من الاتجاه والمقدار للمتجه الممتد من
 النقطة O إلى النقطة P.
 
 
 يُعتبر متجه الموضع متجهًا يمتد من النقطة A إلى النقطة B في الفضاء، ويمكن الإشارة إليه بالرمز
 r. كما يمكن التعبير عن هذا المتجه باستخدام رمزين تحتيين للإشارة إلى النقاط التي يمتد بينها، بحيث
 يُكتب على الشكل rAB. يرجى ملاحظة أنه إذا كان متجه الموضع ممتدًا من نقطة أصل الإحداثيات، فإنه
 يُشار إليه باستخدام رمز تحتي واحد فقط، مثل rA او rB
 يمكن حساب متجه الموضع rAB باستخدام متجهي الموضع rA و
 rB وفقًا للعلاقة التالية:rAB=rB-rA=(xB-xA)i+(yB-yA)j+(zB-zA)k
 

 
 على سبيل المثال، لإنشاء متجه الموضع من النقطة A إلى النقطة B، يتم طرح إحداثيات نقطة الذيل
 A(1 m, 0 m, -3 m) من إحداثيات نقطة الرأس B(-2 m, 2 m, 3 m) مما يؤدي إلى:
 rAB=(-2-1)i+(2-0)j+(3-(-3))kوبالتالي، يكون متجه الموضع:rAB=(-3i+2j+6k) m
 

</div>

متجه الموضع هو مفهوم أساسي في الرياضيات يساعد في تحديد موضع نقطة واحدة فيما يتعلق بنقطة أخرى في الفضاء. إنه ...

Education

JoVE Core

Mechanical Engineering

الهندسة الميكانيكية

Unit 1

الوحدة الأولى

Force Vectors

متجهات القوة

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الأساسية

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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Scalar and vector quantities are two main ways to describe physical quantities in mechanics. Common examples include force, speed, velocity, and work. Some of these are scalars, and others are vectors. A scalar is a quantity that has magnitude only. It does not need direction. Mass, area, and length are examples of scalar quantities. Scalars with the same units can be added or subtracted like ordinary numbers. A class that ends 10 min earlier than 50 min lasts 50 min minus 10 min, which is 40...

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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,...

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Vector Orientation with Direction Cosines

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three-dimensional force system:problem solving

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Resolving Forces in 3D Coordinates

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position vectors

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dot product

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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

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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