Relative Position - Physics

1. Fundamental Concepts

Definition: Relative position refers to the location of an object with respect to another object or a reference point.
Reference Frame: A system used to describe the position and motion of objects.
Position Vector: A vector that indicates the position of a point in space relative to a reference point.

2. Key Concepts

Basic Rule: $${\text{{Relative Position}}} = {\text{{Position Vector}}}_{{\text{{Object}}}} - {\text{{Position Vector}}}_{{\text{{Reference Point}}}}$$

Vector Addition: $${\vec{{r}}}_{{\text{{AB}}}} = {\vec{{r}}}_{{\text{{A}}}} + {\vec{{r}}}_{{\text{{B}}}}$$

Application: Used to determine distances and directions between objects in physics problems

3. Examples

Example 1 (Basic)

Problem: Given two points, A at ${\vec{{r}}}_{{\text{{A}}}} = (3, 4)$ and B at ${\vec{{r}}}_{{\text{{B}}}} = (7, 8)$, find the relative position vector ${\vec{{r}}}_{{\text{{AB}}}}$.

Step-by-Step Solution:

Calculate the difference in coordinates: ${\vec{{r}}}_{{\text{{AB}}}} = (7 - 3, 8 - 4)$
Simplify: ${\vec{{r}}}_{{\text{{AB}}}} = (4, 4)$

Validation: Substitute values → Original: (3, 4) and (7, 8); Simplified: (4, 4) ✓

Example 2 (Intermediate)

Problem: If the position vectors of points P and Q are ${\vec{{r}}}_{{\text{{P}}}} = (2, 5, 3)$ and ${\vec{{r}}}_{{\text{{Q}}}} = (-1, 2, 6)$, find the relative position vector ${\vec{{r}}}_{{\text{{PQ}}}}$.

Step-by-Step Solution:

Calculate the difference in coordinates: ${\vec{{r}}}_{{\text{{PQ}}}} = (-1 - 2, 2 - 5, 6 - 3)$
Simplify: ${\vec{{r}}}_{{\text{{PQ}}}} = (-3, -3, 3)$

Validation: Substitute values → Original: (2, 5, 3) and (-1, 2, 6); Simplified: (-3, -3, 3) ✓

4. Problem-Solving Techniques

Visual Strategy: Use diagrams to represent positions and vectors visually.
Error-Proofing: Double-check coordinate differences by re-calculating them from different perspectives.
Concept Reinforcement: Practice with various reference frames to understand relative positioning better.