Introduction
Relative Movement Problems involve two or more people/objects moving simultaneously or sequentially in different directions. The goal is to determine their relative positions, directions, distances between them, or who is to the left/right of whom after movements. These questions appear frequently in reasoning tests and require careful vector thinking and relative-frame reasoning.
This pattern is important because it trains you to compare multiple trajectories, use cancellations, and reason about positions from different reference points - skills that improve spatial intuition and exam performance.
Pattern: Relative Movement Problems
Pattern
Key concept: Treat each person's path as a vector (horizontal and vertical components). Compute final coordinates relative to a common origin, then compare coordinates to answer relative-position questions (e.g., who is north/east of whom, distances, or facing relations).
Quick rules:
- Choose a convenient origin (often one person's start point) and use consistent axes: North = +y, South = -y, East = +x, West = -x.
- Translate each movement into x/y changes and sum them to get final coordinates.
- Compare coordinates pairwise: x larger → more east; y larger → more north.
- Distance between two points = √((Δx)² + (Δy)²).
- For relative facing questions, track orientations separately from positions; turns affect facing, not coordinates.
Step-by-Step Example
Question
How to solve: Two friends A and B start at the same point. A walks 4 m North then 6 m East. B walks 3 m East then 5 m North. After these moves, who is farther East and what is the distance between them?
Solution
-
Step 1: Choose origin & axes
Place the common start at (0,0). Use x for East (+), y for North (+). -
Step 2: Convert A’s movements to coordinates
A: 4 m North → (0, +4). Then 6 m East → (0+6, 4) = (6, 4). -
Step 3: Convert B’s movements to coordinates
B: 3 m East → (3, 0). Then 5 m North → (3, 5) = (3, 5). -
Step 4: Compare East (x) coordinates
A’s x = 6, B’s x = 3 → A is farther East. -
Step 5: Distance between A and B
Δx = 6 - 3 = 3; Δy = 4 - 5 = -1. Distance = √(3² + (-1)²) = √(9 + 1) = √10 ≈ 3.16 m. -
Final Answer:
A is farther East, and the distance between them is √10 ≈ 3.16 m. -
Quick Check:
Compare coordinates visually: A(6,4) right of B(3,5) by 3 units and 1 unit south → diagonal distance √10 ✅
Quick Variations
1. Simultaneous movement in different starting points - translate both to global coordinates first.
2. Questions asking “who faces whom” - compute final positions then reason about facing using relative bearings.
3. Some problems involve return paths or loops; treat each leg sequentially and sum vectors.
4. Time-staggered movement (A moves, then B starts) - still compute final coordinates at the relevant times for comparison.
5. Combine with turns/rotations: track both position (vector) and facing (orientation) separately.
Trick to Always Use
- Step 1: Fix an origin and axis convention (x = East, y = North).
- Step 2: Convert every move into (Δx, Δy); add them sequentially per person.
- Step 3: For comparisons: larger x → more east; larger y → more north. For distance, use Pythagoras.
- Step 4: Draw a tiny diagram with labeled coordinates - it catches sign errors quickly.
Summary
Summary
For Relative Movement Problems:
- Always convert movements to coordinate changes.
- Compute final positions separately for each person/object.
- Compare coordinates to answer relative-location questions; use √(Δx²+Δy²) for distances.
- Keep position (vector) and facing (orientation) tracked separately when turns are involved.
- Quick diagram + axis convention prevents sign mistakes and speeds up solving.
Hint: Subtract B’s coordinates from A’s to find direction; use √(Δx² + Δy²) for exact distance.
Common Mistakes: Labeling as '2 m' instead of √8 m, or reversing A-B reference.