Direction and Distance — Study Notes

Overview

Direction and Distance problems form a staple of the SSC CHSL General Intelligence section, appearing in 2–3 questions every year. These questions test your ability to visualize movement in two dimensions using compass directions (North, South, East, West and their combinations) and calculate distances using basic geometry.

Mastery requires two skills: accurate mental tracking of position changes as someone moves in multiple steps, and applying the Pythagorean theorem to find shortest distances. Most errors come from confusing left-right turns or mixing up direction conventions. The good news—once you internalize the compass rose and a few standard patterns, these become scoring questions in under 60 seconds.

Expect three question types: (1) finding the final direction from the starting point, (2) calculating the shortest (straight-line) distance between start and end points, and (3) determining the direction of one point relative to another after a series of movements. All require careful diagram sketching, which is your primary tool.

Key Concepts

• **Compass conventions**: North is always "up," South "down," East "right," West "left" on paper. Northeast lies 45° between North and East; similar for NW, SE, SW. Never assume left/right means compass direction—it's relative to the person's current facing.

• **Displacement vs. distance**: Distance is the total path length traveled; displacement (shortest distance) is the straight-line distance from start to finish. Always use Pythagorean theorem for displacement: d = √(x² + y²) where x and y are net horizontal and vertical movements.

• **Coordinate method**: Treat the starting point as origin (0,0). Assign +y for North, –y for South, +x for East, –x for West. After all movements, sum x-components and y-components separately to find final coordinates, then calculate distance.

• **Turn interpretation**: "Turns left" or "turns right" depends on current facing direction. If facing North and turning right, you face East. If facing South and turning right, you face West. Draw an arrow for the person's facing direction at each step.

• **Final direction from start**: After plotting final position (x_f, y_f), determine quadrant: x_f > 0, y_f > 0 → NE; x_f < 0, y_f > 0 → NW; x_f < 0, y_f < 0 → SW; x_f > 0, y_f < 0 → SE. Exact compass direction uses tan(θ) = y_f / x_f if needed, but exams rarely require angle calculation.

• **Shadow-based direction** (bonus type): Morning shadow falls West (sun in East), evening shadow falls East. At noon, shadow points North in Northern Hemisphere. Use shadow direction to infer person's facing or position.

• **Common movements**: "Moves towards" means traveling in that direction. "Faces towards" means pointing body that way without movement. Don't confuse the two.

Formulas / Key Facts

1. **Pythagorean theorem**: Shortest distance = √(horizontal displacement² + vertical displacement²)

2. **Net displacement components**: x_net = Σ(Eastward movements) – Σ(Westward movements); y_net = Σ(Northward movements) – Σ(Southward movements)

3. **Right-turn sequence**: North → East → South → West → North (clockwise)

4. **Left-turn sequence**: North → West → South → East → North (anticlockwise)

5. **90° turn**: Changes facing by one cardinal direction (N/E/S/W). 180° turn reverses facing direction. 45° turn moves to intercardinal (NE/SE/SW/NW).

6. **Opposite directions**: North ↔ South, East ↔ West, NE ↔ SW, NW ↔ SE

7. **Standard distance for 45° movement**: If moving NE/NW/SE/SW for distance d, horizontal and vertical components are each d/√2 ≈ 0.707d. Most exams keep it simple with perpendicular movements.

8. **Shadow rules**: Morning (6–12): shadow → West. Evening (12–6): shadow → East. Noon: shadow → North (shortest).

Worked Examples

**Example 1**: A man walks 5 km North, then turns right and walks 12 km. What is the shortest distance from his starting point?

**Solution**:

• Starting point: (0, 0)
• After 5 km North: (0, 5)
• Turns right (now facing East), walks 12 km East: (12, 5)
• Shortest distance = √(12² + 5²) = √(144 + 25) = √169 = **13 km**

**Example 2**: A girl starts from home, walks 40 m towards South, turns left and walks 30 m, then turns right and walks 20 m. In which direction is she from her starting point?

**Solution**:

• Start: (0, 0), facing reference not given, so track position only
• 40 m South: (0, –40)
• Turns left (from South = faces East), walks 30 m: (30, –40)
• Turns right (from East = faces South), walks 20 m: (30, –60)
• Final position: x > 0 (East), y < 0 (South) → **South-East**

**Example 3**: Rahul walks 10 km West, then 10 km North, then 10 km East. How far is he from the starting point and in which direction?

**Solution**:

• Start: (0, 0)
• 10 km West: (–10, 0)
• 10 km North: (–10, 10)
• 10 km East: (0, 10)
• Net displacement: x = 0, y = 10
• Distance = √(0² + 10²) = **10 km**
• Direction from start: directly **North**

Common Mistakes

1. **Confusing left/right with compass directions**: "Turns left" is relative to current facing, not absolute. If facing South, turning left means facing East, not West. Always track the facing direction separately.

2. **Adding distances instead of components**: Students add 3 km N + 4 km E = 7 km for shortest distance. Wrong—use Pythagoras: √(3² + 4²) = 5 km. Never sum path lengths for displacement.

3. **Sign errors in coordinate method**: Forgetting that South is negative y and West is negative x. Write South = –y and West = –x at the top of your rough work every time.

4. **Misjudging final direction quadrant**: Position (–5, 3) is North-West, not North-East. Negative x always means West component. Double-check signs before declaring direction.

5. **Ignoring "turns to face"**: When a question says "turns to face North," it's resetting facing direction, not indicating movement. Mark facing with an arrow separate from position dot.

Quick Reference

• **Coordinate signs**: North = +y, South = –y, East = +x, West = –x
• **Shortest distance formula**: √(x² + y²) where x, y are net horizontal and vertical displacements
• **Right turn from North**: N → E → S → W → N (clockwise)
• **Direction quadrants**: (+x, +y) = NE, (–x, +y) = NW, (–x, –y) = SW, (+x, –y) = SE
• **Shadow direction**: Morning shadow → West, Evening shadow → East
• **Always draw a diagram**: Mark start point, plot each movement with arrows, label coordinates—never solve mentally for multi-step problems