🧭 Direction Sense - Complete Theory

Master directional navigation - the most calculation-based reasoning topic!

🎯 What is Direction Sense?

Direction Sense questions test your ability to:

• Track movement in different directions
• Calculate final position from starting point
• Find shortest distance using Pythagoras theorem
• Understand compass directions and turns

Example:

A person walks 5 km North, then 3 km East. What is the shortest distance from starting point?

Answer: √(5² + 3²) = √34 km

📐 Basic Directions

Cardinal Directions (Main)

          North (N)
             ↑
             |
             |

West (W) ←——-+——-→ East (E) | | ↓ South (S)

Key Points:

• North ↔ South (Opposite)
• East ↔ West (Opposite)
• 4 main directions = Cardinal directions

Ordinal Directions (Diagonal)

    NW      N      NE
      \     |     /
       \    |    /
        \   |   /
    W ----  +  ---- E
        /   |   \
       /    |    \
      /     |     \
    SW      S      SE

8 Total Directions:

1. North (N)
2. North-East (NE)
3. East (E)
4. South-East (SE)
5. South (S)
6. South-West (SW)
7. West (W)
8. North-West (NW)

🔄 Turns and Rotations

Right Turn (Clockwise)

Starting from NORTH, turning RIGHT:

N → E → S → W → N (360° cycle)

Each right turn = 90° clockwise

Examples:

Facing North + Right turn = Facing East Facing East + Right turn = Facing South Facing South + Right turn = Facing West Facing West + Right turn = Facing North

Left Turn (Anti-clockwise)

Starting from NORTH, turning LEFT:

N → W → S → E → N (360° cycle)

Each left turn = 90° anti-clockwise

Examples:

Facing North + Left turn = Facing West Facing West + Left turn = Facing South Facing South + Left turn = Facing East Facing East + Left turn = Facing North

About Turn (180°)

About turn = 2 consecutive right turns OR 2 consecutive left turns

Facing North + About turn = Facing South Facing East + About turn = Facing West Facing South + About turn = Facing North Facing West + About turn = Facing East

📊 Angle-Based Turns

Standard Angles

Right angle = 90° Straight angle = 180° (About turn) Complete rotation = 360°

Calculating Final Direction

From North:

• 90° clockwise = East
• 180° clockwise = South
• 270° clockwise = West
• 360° clockwise = North (back to start)

From North:

• 45° clockwise = North-East
• 135° clockwise = South-East
• 225° clockwise = South-West
• 315° clockwise = North-West

💡 Solved Examples

Example 1: Basic Movement

Q: A man walks 3 km North, then turns right and walks 4 km. How far is he from the starting point?

Solution:

Step 1: Draw diagram

    Final (B)
      |
      | 4 km (East)
      |

Start —-+ (A) 3 km (North)

Movement: A → 3 km North → Point P P → Turn right (now facing East) → 4 km East → Point B

Step 2: Calculate shortest distance (A to B)

Forms a right triangle:

• One side (North) = 3 km
• Other side (East) = 4 km

Using Pythagoras: Distance² = 3² + 4² Distance² = 9 + 16 = 25 Distance = 5 km

Answer: 5 km

Example 2: Multiple Turns

Q: A person starts facing North. He turns 90° right, then 180° left, then 90° right. Which direction is he facing now?

Solution:

Step 1: Track each turn

Start: Facing North (N)

Turn 1: 90° right N → E (Facing East)

Turn 2: 180° left E → 180° anti-clockwise E → N → W (Facing West)

Turn 3: 90° right W → 90° clockwise W → N (Facing North)

Answer: North

Example 3: Distance Calculation

Q: Ram walks 5 km East, then 5 km North, then 5 km West. What is the shortest distance from starting point?

Solution:

Step 1: Draw path

  C ← 5 km W ← B
  |            ↑
  |            | 5 km N
  |            |
  Start → 5 km E → A

Step 2: Simplify

Net East-West displacement: 5 km East - 5 km West = 0 km (canceled out)

Net North-South displacement: 5 km North = 5 km North

Step 3: Final position

Ram is 5 km North of starting point Shortest distance = 5 km

Answer: 5 km North (or just 5 km)

Example 4: Diagonal Movement

Q: A person walks 10 km North-East. What is his displacement in North and East directions?

Solution:

Step 1: Understand North-East

North-East = 45° angle between North and East

    NE (10 km)
   /|
  / |
 /  | North component
/   |

/45° | /_____| East component

Step 2: Calculate components

For 45° angle: North component = 10 × cos(45°) = 10 × (1/√2) = 10/√2 km East component = 10 × sin(45°) = 10 × (1/√2) = 10/√2 km

Both are equal = 10/√2 = 5√2 ≈ 7.07 km

Answer: 7.07 km North and 7.07 km East

Example 5: Complex Path

Q: Starting from home, Amit walks 40 m North, then 30 m East, then 40 m South, then 20 m West. How far is he from home?

Solution:

Step 1: Draw and track

       B (30m E)
       |
       |

Home ——A (40m N) | | | C (40m S) |__________| D (20m W)

Net North-South: 40 m North - 40 m South = 0 m

Net East-West: 30 m East - 20 m West = 10 m East

Step 2: Final position

Amit is 10 m East of home Distance = 10 m

Answer: 10 m

Example 6: Shadow Problem

Q: One morning, Rajiv was facing a pole. The shadow of the pole fell exactly to his right. Which direction was he facing?

Solution:

Step 1: Understand shadow direction

Morning: Sun is in the EAST Shadow falls opposite to sun = WEST direction

Step 2: Determine facing

Shadow fell to his RIGHT = Shadow in WEST If West is to the right, then: He must be facing NORTH

Verification: Facing North → Right = East? NO Facing North → Right = West? NO

Wait! Let’s reconsider: Facing direction + Right side = West

If facing North: Right = East ✗ If facing South: Right = West ✓

Answer: South

⚡ Quick Formulas & Shortcuts

Formula 1: Pythagoras Theorem

For right-angled triangle: Shortest distance = √(a² + b²)

Common Pythagorean triplets (memorize!): 3-4-5 (3² + 4² = 5²) 5-12-13 8-15-17 7-24-25

Formula 2: Net Displacement

Net North-South = (Total North) - (Total South) Net East-West = (Total East) - (Total West)

Final distance = √[(Net N-S)² + (Net E-W)²]

Formula 3: Turn Counting

Total right turns (R), Total left turns (L)

Net turns = |R - L|

If R > L: Net right turns = R - L If L > R: Net left turns = L - R

Formula 4: Shadow Direction

Morning (before noon): Sun in EAST, Shadow in WEST Evening (after noon): Sun in WEST, Shadow in EAST

If shadow to your LEFT:

• Morning: Facing SOUTH
• Evening: Facing NORTH

If shadow to your RIGHT:

• Morning: Facing NORTH
• Evening: Facing SOUTH

📊 Direction Table

Right Turn Sequence (Clockwise)

Left Turn Sequence (Anti-clockwise)

⚠️ Common Mistakes

❌ Mistake 1: Left-Right Confusion

Wrong: Right turn from North = West ✗ Right: Right turn from North = East ✓

Remember: RIGHT = CLOCKWISE (N → E → S → W)

❌ Mistake 2: Shadow Direction

Wrong: Morning shadow always falls East ✗ Right: Morning shadow falls WEST (opposite to sun) ✓

❌ Mistake 3: Forgetting to Square Root

Wrong: Distance = 3² + 4² = 25 km ✗ Right: Distance = √(3² + 4²) = √25 = 5 km ✓

❌ Mistake 4: Not Simplifying First

Wrong: Calculate √(40² + 30²) directly ✗ Right: Simplify to √(4² × 10² + 3² × 10²) = 10√(16+9) = 50 km ✓

❌ Mistake 5: Diagonal as Straight

Wrong: 10 km NE = 10 km North + 10 km East ✗ Right: 10 km NE = 7.07 km North + 7.07 km East ✓

🎯 Advanced Patterns

Pattern 1: Minimum Distance Between Two Movers

Q: A and B start from the same point. A goes 5 km North, B goes 5 km East. What is distance between them?

Solution:

A
|
| 5 km
|

Start —- 5 km —- B

Distance AB = √(5² + 5²) = √50 = 5√2 km

Pattern 2: Meeting Point Problems

Q: A walks 3 km North. B walks 4 km East from the same starting point. If they walk towards each other in straight line, where do they meet?

Solution:

They meet at a point on the straight line connecting them. The straight line distance = √(3² + 4²) = 5 km

Pattern 3: Circular Path

Q: A person walks in a square path: 10m North, 10m East, 10m South, 10m West. What is displacement?

Solution:

 10m E

+——+ | | 10m N | 10m S | | +——+ 10m W

Net displacement = 0 (back to starting point)

📝 Practice Problems

Level 1: Basic Direction

1. A man walks 8 km South, then 6 km East. What is the shortest distance from start?

2. Facing North, a person turns right, then right again. Which direction is he facing?

3. The shadow of a pole falls to the North in the evening. Where is the sun?

Level 2: Medium

4. Ram walks 10 km North, 6 km East, 10 km South, 6 km West. How far from starting point?

5. A person walks 7 km East, then turns 135° clockwise and walks 7 km. Find final direction.

6. A faces North. After 3 right turns and 2 left turns, which direction does he face?

Level 3: Hard

7. A walks 15 km North, then turns right and walks 20 km, then turns right and walks 15 km. Final distance from start?

8. One morning, Suresh was facing a tree. The shadow of the tree fell exactly behind him. Which direction was he facing?

9. A walks 12 km North-West. What is his displacement in North and West directions separately?

🎯 Special Cases

Case 1: 3-4-5 Triangle (Most Common)

If distances are multiples of 3 and 4: Answer is multiple of 5

Example: 6 km North + 8 km East = 10 km (2 × 5) 9 km North + 12 km East = 15 km (3 × 5)

Case 2: Equal Perpendicular Distances

If both distances are equal: Answer = Distance × √2

Example: 5 km North + 5 km East = 5√2 km ≈ 7.07 km 10 km North + 10 km West = 10√2 km ≈ 14.14 km

Case 3: Opposite Direction Cancellation

If movement in opposite directions: They cancel out

Example: 10 km North + 7 km South = Net 3 km North 15 km East + 15 km West = Net 0 km

🎯 Exam Strategy

Time Management:

• Per question: 45-60 seconds
• For 5 direction questions: 4-5 minutes

Quick Approach:

1. Draw rough diagram (15 sec) - Always!
2. Track movements (15 sec)
3. Calculate net displacement (15 sec)
4. Apply Pythagoras (10 sec)
5. Verify (5 sec)

Priority:

• ✅ Simple turn questions - 25 sec
• ✅ Distance with 3-4-5 triplets - 35 sec
• ✅ Shadow problems - 30 sec
• ⏭️ Complex multi-step paths - 60+ sec

🔗 Related Topics

Uses Concepts From:

• Basic geometry (Pythagoras theorem)
• Angle measurement
• Coordinate geometry

Related Reasoning Topics:

• Puzzles - Position mapping
• Blood Relations - Relationship tracking

Practice:

• Direction Sense Questions

Master Direction Sense - Draw diagrams, use Pythagoras! 🧭