Direction & Distance

Core Formulas

• Standard compass: N, S, E, W (and NE, NW, SE, SW between them)
• Right turn from facing North → now facing East; Left turn from North → now facing West (rotate the compass 90° in the turn direction)
• Shortest/final distance from start (not total path walked) uses the straight-line (resultant) distance:

$$d=\sqrt{x^2+y^2}$$

where x and y are the net displacement along two perpendicular axes

Turn Rules Quick Reference

• Facing North: Right turn → East; Left turn → West
• Facing East: Right turn → South; Left turn → North
• Facing South: Right turn → West; Left turn → East
• Facing West: Right turn → North; Left turn → South
• A 180° turn (either direction) always means facing the exact opposite direction

Worked Examples

1. Walk 3 km North, then 4 km East → shortest distance back:

$$\sqrt{3^2+4^2}=5 \text{ km}$$

1. Walk 6 km South, then 8 km West → shortest distance to start:

$$\sqrt{6^2+8^2}=10 \text{ km}$$

1. A man walks 5 km North, turns right, walks 5 km, turns right again, walks 5 km. What direction is he from the start, and how far?

   → North 5km → turn right (facing East) walk 5km → turn right (facing South) walk 5km → the N and S legs cancel → he is 5 km East of start

2. Facing North, a person turns 90° clockwise, then 180°, then 90° anticlockwise. Which direction do they face now?

   → North → (90° CW) East → (180°) West → (90° CCW) South

Drill

1. Walk 12 km East, then 5 km North. Shortest distance to start? — 13 km (5-12-13 triple)
2. Facing South, turn left twice (90° each). Which direction now? — North
3. A person walks 4km North, 3km East, 4km South. Final position relative to start? — 3 km East
4. Facing East, turn right, then right again. Which direction? — West
5. Walk 8km North, 6km East, 8km South, 6km West. Distance from start? — 0 km (back at start)

Practice tracker

YYYY-MM-DD

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