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

וקטור מיקום הוא וקטור קבוע הממקם את מיקומה של נקודה במרחב ביחס לנקודה אחרת.
חשבו על עין בורג המקובעת לקיר וכבל OP מחובר אליה. הקמת מערכת קואורדינטות קרטזית עם עין בורג במקור ווקטורי הרכיבים לאורך הצירים כדי לאתר את נקודה P על הכבל.
וקטור מיקום r המכוון מהמוצא לנקודה P יכול לבוא לידי ביטוי בצורה וקטורית קרטזית.
לייצוג כללי של וקטור קרטזי, שקול סוגר קבוע על קיר וכבל המחובר אליו.
נניחש-n A ו-n B יהיו וקטורי המיקום המשתרעים מהמוצא לנקודות A ו-B.
כך שניתן לקבל את וקטור המיקום המתקבל n מנקודה A לנקודה B, באמצעות כלל המשולש ועל ידי ביטוי nA ו- nB בצורה וקטורית קרטזית.
i, j ו-k הם וקטורי היחידה לאורך מרכיבי וקטור המיקום n הנוצרים על ידי חיסור הקואורדינטות של A מ-B.

<ol>
	<li>ראסל היבלר. (2016). מכניקה הנדסית: סטטיקה ודינמיקה. פירסון. (עמ' לא – 56,57)</li>
</ol>

וקטורי מיקום

Position Vectors

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וקטור מיקום הוא מושג בסיסי במתמטיקה שעוזר לקבוע את מיקומה של נקודה אחת ביחס לנקודה אחרת במרחב. זהו וקטור המתאר את הכיוון והמרחק בין שתי נקודות. וקטורי מיקום שימושיים מאוד בתחום המתמטיקה והמדעים, מכיוון שהם עוזרים לייצג יחסים מרחביים ומקלים על החישובים. 


לדוגמה, אנו רוצים לאתר נקודה P(x, y, z) ביחס למקור הקואורדינטות O. במקרה זה, נוכל להגדיר וקטור מיקום r, המשתרע מהמקור O לנקודה P. אנו יכולים לבטא את הווקטור הזה בצורה וקטורית קרטזית כמו: r = xi + yj + zk, כאשר i, j ו-k הם וקטורי היחידה בכיווני-x, y, ו-z בהתאמה. וקטור המיקום r נותן לנו את הכיוון והגודל של הווקטור מנקודה O לנקודה P.


חשבו על וקטור מיקום המכוון מנקודה A לנקודה B במרחב. וקטור זה יכול להיות מסומן על ידי הסמל r. אנו יכולים גם להתייחס לוקטור זה עם שני כתוביות כדי לציין את הנקודות מהן ואליהן הוא מכוון. לפיכך, אנו יכולים גם לייעד את r כrAB. שים לב שאם וקטורי המיקום משתרעים ממקור הקואורדינטות, אזי הם מכונים רק עם סימון אחד, כrA וr ב. וקטור המיקום rAB ניתן לקבל מrA וrB באמצעות הביטוי rAB = rB - rA= (xB - xA)i + (yB - yA)j + (zB - zA)k.


לדוגמה, כדי לקבוע וקטור מיקום מנקודה A עד B, הקואורדינטות של הזנב A(1 m, m0, -3 מ') מופחתים מהקואורדינטות של הראש B(-2 מ', 2 מ', 3 מ'), מה שגורם לrAB ={ -3i + 2j + 6k} מ.
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וקטור מיקום הוא מושג בסיסי במתמטיקה המסייע לקבוע את מיקומה של נקודה אחת ביחס לנקודה אחרת במרחב. זה ...

Education

JoVE Core

Mechanical Engineering

Unit 1

Force Vectors

KEY TERMS AND CONCEPTS

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.

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Using Position Vectors to Locate Points

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