Direction problems test whether we can mentally trace a path and figure out where someone ends up. The trick is dead simple: draw the path on paper. No matter how complex the turns and distances, if we sketch it step by step, the answer becomes obvious. Let’s never try to solve these in our heads.
The 8 Cardinal Directions
Memory trick for turns: Imagine we’re standing and facing North. Turn right → we face East. Turn right again → South. Right again → West. Right again → North. So a right turn goes N→E→S→W→N (clockwise). Left turn is the reverse: N→W→S→E→N (counter-clockwise).
The Path-Drawing Method
This is our main approach. For every direction problem:
- Draw a small compass (N/S/E/W) in the corner of our rough work
- Mark the starting point
- Draw each movement as a straight line in the correct direction
- Label distances on each segment
- Connect start to end for the shortest distance (if asked)
Example 1: Basic path tracing
A person walks 5 km North, then 3 km East, then 5 km South. How far is he from the starting point and in which direction?
Let’s trace the path:
- Start at point O, walk 5 km North to point A
- From A, walk 3 km East to point B
- From B, walk 5 km South to point C
The North (5 km) and South (5 km) cancel each other out. The only net displacement is 3 km East.
Answer: 3 km East of the starting point.
Example 2: Shortest distance with Pythagoras
Ravi starts from home, walks 4 km North, then turns right and walks 3 km. What is the shortest distance from home?
Path:
- Start at O, walk 4 km North to A
- Turn right from North = face East, walk 3 km to B
The path forms a right angle. The shortest distance (O to B) is the hypotenuse.
Shortest distance = √(4² + 3²) = √(16 + 9) = √25 = 5 km
Direction from home: North-East (since B is both north and east of O).
Example 3: Multiple turns
A man walks 3 km East, turns left, walks 4 km, turns left, walks 3 km, turns left, walks 2 km. How far is he from the starting point?
Let’s trace:
- Start at O, walk 3 km East to A
- Turn left from East = face North, walk 4 km to B
- Turn left from North = face West, walk 3 km to C
- Turn left from West = face South, walk 2 km to D
Now let’s figure out position of D relative to O:
- East-West: went 3 km East, then 3 km West → net = 0 km (back on the same vertical line)
- North-South: went 4 km North, then 2 km South → net = 2 km North
Distance from start = 2 km North
Example 4: Diagonal movement
Priya walks 6 km towards South, then turns left and walks 4 km, then turns left and walks 3 km. How far and in which direction is she from the starting point?
Trace:
- Start at O, walk 6 km South to A
- Turn left from South = face East, walk 4 km to B
- Turn left from East = face North, walk 3 km to C
Net displacement:
- North-South: 6 km South - 3 km North = 3 km South
- East-West: 4 km East
Shortest distance = √(3² + 4²) = √(9 + 16) = √25 = 5 km
Direction: She is South-East of the starting point (both south and east).
Shadow-Based Direction Problems
These combine direction sense with the sun’s position. The key facts are:
- Sun rises in the East (morning)
- Sun sets in the West (evening)
- Shadow falls opposite to the sun’s direction
- Morning: Sun is in East → shadow falls towards West
- Evening: Sun is in West → shadow falls towards East
- Noon: Sun is overhead → shadow is directly below (very short, towards South in Northern hemisphere)
Example 5: Shadow problem
One morning, Suresh was walking. His shadow fell to his right. Which direction was he facing?
Morning → Sun is in the East → Shadow falls to the West.
If the shadow is to his right, then West is to his right. If West is to our right, we must be facing South.
Think of it this way: Stand facing South. East is to our left, West is to our right. Morning shadow goes West = to the right. ✓
Answer: South
Example: Evening shadow
In the evening, Priya noticed her shadow fell to her left. Which direction was she facing?
Evening → Sun in the West → Shadow falls to the East.
Shadow is to her left, so East is to her left. If East is to our left, we’re facing North.
Answer: North
Turn-Based Direction Change
Sometimes we get turns expressed in degrees:
- 90° right turn = standard right turn (N→E, E→S, S→W, W→N)
- 90° left turn = standard left turn (N→W, W→S, S→E, E→N)
- 180° turn = about-face (N→S, E→W, etc.)
- 45° right from North = North-East
- 135° right from North = South-East (90° + 45°)
Example: Degree-based turns
A person facing North turns 90° clockwise, then 135° anti-clockwise. Which direction is he facing now?
- Facing North, turn 90° clockwise = facing East
- From East, turn 135° anti-clockwise:
- 90° anti-clockwise from East = North
- 45° more anti-clockwise = North-West
Answer: North-West
Shortcut Tips
- Always draw the path. Even if the problem seems simple. It takes 10 seconds and saves us from silly mistakes.
- Cancel opposite movements. North cancels South, East cancels West. Find the net displacement in each axis.
- For shortest distance, use Pythagoras: √(net_horizontal² + net_vertical²).
- For shadow problems, first determine sun position (morning=East, evening=West), then shadow is opposite.
- “Turn right” from any direction = clockwise 90°. “Turn left” = counter-clockwise 90°.
Common Exam Variations
- Find the shortest distance from start to end
- Find the direction of final position relative to start
- Shadow-based direction determination
- Clock + direction hybrid (facing the direction of 7:30 on a clock, etc.)
- Multiple people starting from different points
- Which direction should the person turn to face a specific point
Practice Problems
Problem 1: A man walks 1 km East, then 2 km North, then 1 km East, then 1 km North. How far is he from the starting point (shortest distance)?
Problem 2: On an evening, Maya was standing in a park. Her shadow fell in front of her. Which direction was she facing?
Problem 3: Starting from point A, Raj walks 5 km South, turns left, walks 3 km, turns left, walks 5 km, turns right, walks 4 km. How far is he from point A and in which direction?
Answers
Problem 1: Net East-West = 1 + 1 = 2 km East. Net North-South = 2 + 1 = 3 km North. Shortest distance = √(2² + 3²) = √(4+9) = √13 ≈ 3.6 km (North-East direction).
Problem 2: Evening → Sun in the West → Shadow falls towards East. If the shadow is in front of her, she is facing East.
Problem 3: Trace: 5 km South → 3 km East (left from South) → 5 km North (left from East) → 4 km East (right from North). Net N-S: 5 South - 5 North = 0. Net E-W: 3 + 4 = 7 km East. He is 7 km East of point A.