A man starts walking from home and walks 4 miles east, 4 miles southeast, 7 miles south, 6 miles southwest, and 2 miles east. How far has he walked? If he walked straight home, how far would he have to walk? (Round your answer to three decimal places
Answer
🚶 Vector Path Analysis – Distance and Displacement
🔍 Step 1: Total Distance Walked
The man walks the following segments:
- 4 miles east
- 4 miles southeast
- 7 miles south
- 6 miles southwest
- 2 miles east
Total distance walked is the sum of all path lengths:
4 + 4 + 7 + 6 + 2 = 23 miles
🧭 Step 2: Resolve Each Segment Into Components
Let’s consider east as positive x and north as positive y.
| Segment | Direction | Distance | X-component | Y-component |
|---|---|---|---|---|
| 1 | East | 4 | +4.000 | 0.000 |
| 2 | Southeast (45°) | 4 | +2.828 | −2.828 |
| 3 | South | 7 | 0.000 | −7.000 |
| 4 | Southwest (45°) | 6 | −4.243 | −4.243 |
| 5 | East | 2 | +2.000 | 0.000 |
🧮 Step 3: Total Displacement Components
X-total = 4.000 + 2.828 + 0 − 4.243 + 2.000 = 4.585
Y-total = 0 − 2.828 − 7.000 − 4.243 + 0 = −14.071
📐 Step 4: Calculate Straight-Line Distance (Displacement)
Use the Pythagorean theorem:
Displacement = √(x² + y²)
= √(4.585² + (−14.071)²)
= √(21.03 + 198.00)
= √219.03 ≈ 14.797 miles
= √(4.585² + (−14.071)²)
= √(21.03 + 198.00)
= √219.03 ≈ 14.797 miles
✅ Final Answers:
- Total distance walked: 23.000 miles
- Straight-line distance to walk back home: 14.797 miles
💡 Tip: Always break angled movements into x and y components when solving displacement problems. Use trigonometric functions like cos(θ) and sin(θ) for precise vector resolution.