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

لنأخذ المتجه A مع مكوناته x و y الممثلة بدلالة متجهات الوحدة i و j. هنا متجهات الوحدة لها حجم بلا أبعاد يساوي واحدا.
نظرا لأن حجم أي مكون متجه هو دائما كمية موجبة ، ممثلة بالمقاييس ، يمكن التعبير عن A كمتجه ديكارتي.
هنا ، يتم استخدام نظام إحداثيات مستطيل باليد اليمنى. يشير الإبهام الأيمن نحو المحور z الموجب ، وتتجعد الأصابع من المحور x الموجب باتجاه المحور y الموجب.
يمكن تمثيل متجه ثلاثي الأبعاد بإحداثيات ديكارتية مستطيلة باستخدام متجهات الوحدة i و j و k. يتم تمثيل اتجاه هذه المتجهات اعتمادا على المحاور الموجبة أو السالبة.
يتم تمثيل المتجه على أنه مجموع المتجه لمكوناته الفردية ، ويتم التعبير عن حجمه على أنه الجذر التربيعي الموجب لمجموع مربعات مكوناته.
يتم تبسيط عمليات الجبر المتجه من خلال تمثيل المتجه في الشكل الديكارتي. يفصل حجمه واتجاهه على طول المحاور باستخدام تدوين متجه الوحدة.

<ol>
	<li>هيبلر ، آر سي (2016). الميكانيكا الهندسية - الإحصاء والديناميكيات. هوبوكين ، نيو جيرسي: قاعة بيرسون برنتيس. ص صفحة رقم 34 ، 44 ، 45.</li>
</ol>

التمثيل البياني للمتجهات الديكارتية

<div style="direction:rtl">
يعتبر التمثيل البياني للمتجهات الديكارتية هو أداة قيمة في الهندسة الميكانيكية لتمثيل المتجهات في الفضاء ثلاثي الأبعاد، وإجراء العمليات المتجهية مثل استنباط التدرج (Gradient)، والتباعد (Divergence)، والانحناء (Curl)، بالإضافة إلى التعبير عن الكميات الفيزيائية مثل الإزاحة، السرعة، التسارع، والقوة. من خلال استخدام التمثيل البياني للمتجهات الديكارتية، يتمكن المهندسون من تحليل المشكلات وحلها بسهولة أكبر في مجالات متعددة من الهندسة الميكانيكية، بما في ذلك الديناميكا، الكينماتيكا، وميكانيكا الموائع. يُمثل هذا النظام المتجهات على شكل ثلاثة مركبات على المحاور x و y و z على التوالي.

على سبيل المثال، لنفترض أن لدينا متجهًا A يشير في اتجاه النقطة (3، -4، 5)، عندها يمكن تمثيله باستخدام التمثيل البياني للمتجهات الديكارتية على النحو التالي:A=3i−4j+5k

حيث i و j و k هي المتجهات الموحدة على المحاور x و y و z على التوالي. يتم تعريف المتجهات الموحدة كما يلي:

i=(1,0,0)،j=(0,1,0)،k=(0,0,1)

يمكن استخدام التمثيل البياني للمتجهات الديكارتية لإجراء عمليات متجهية متعددة، مثل الجمع، الطرح، والضرب القياسي. على سبيل المثال، إذا كان لدينا المتجهان:

A = 3i - 4j + 5k B = 2i + 7j - 3k

فإن جمعهما باستخدام التمثيل البياني للمتجهات الديكارتية يتم كما يلي: <img alt="Equation 1" src="/files/ftp_upload/14232/14232_ Equation_1v2.svg" style="text-align: center; height: 13px;"> وهكذا، تتيح هذه الطريقة التعامل مع المتجهات بدقة ووضوح عند تحليل النظم الهندسية والميكانيكية. يمكننا أيضًا طرحهما على النحو التالي: <img alt="Equation 2" src="/files/ftp_upload/14232/14232_ Equation_2v2.svg" style="text-align: center; height: 13px;"></div>

يعد تدوين المتجه الديكارتي أداة قيمة في الهندسة الميكانيكية لتمثيل المتجهات في الفضاء ثلاثي الأبعاد ، وإجراء عمليات المتجهات ...

Education

JoVE Core

Mechanical Engineering

الهندسة الميكانيكية

Unit 1

الوحدة الأولى

Force Vectors

متجهات القوة

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الأساسية

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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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,...

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three-dimensional force system:problem solving

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Resolving Forces in 3D Coordinates

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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 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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