Episode 8 — Aptitude and Reasoning / 8.20 — Direction Sense
8.20.a Concepts and Formulas -- Direction Sense
1. The Eight Directions
There are 4 cardinal and 4 intercardinal (ordinal) directions:
N (North)
|
|
NW -----------+----------- NE
(North-West) | (North-East)
|
W ----------------+---------------- E
(West) | (East)
|
SW -----------+----------- SE
(South-West) | (South-East)
|
S (South)
Key Points
- North and South are opposite directions.
- East and West are opposite directions.
- NE and SW are opposite; NW and SE are opposite.
- The angle between any two adjacent cardinal directions is 90 degrees.
- The angle between a cardinal and its adjacent intercardinal direction is 45 degrees.
2. Left and Right Turns from Each Direction
Fundamental Rule
- Right turn = Clockwise rotation by 90 degrees
- Left turn = Counter-clockwise rotation by 90 degrees
- About turn / U-turn = Rotation by 180 degrees (reverse direction)
Complete Turn Table
| Facing Direction | Turn Right (90 deg CW) | Turn Left (90 deg CCW) | About Turn (180 deg) |
|---|---|---|---|
| North | East | West | South |
| East | South | North | West |
| South | West | East | North |
| West | North | South | East |
| NE | SE | NW | SW |
| SE | SW | NE | NW |
| SW | NW | SE | NE |
| NW | NE | SW | SE |
Mnemonic: "NEWS Clockwise"
Think of the directions going clockwise: N -> E -> S -> W -> N
- Turning right = next direction in the clockwise sequence
- Turning left = previous direction (or next in counter-clockwise: N -> W -> S -> E)
3. Clockwise Direction Sequence
N
/ \
NW NE
| |
W E Clockwise: N -> NE -> E -> SE -> S -> SW -> W -> NW -> N
| |
SW SE
\ /
S
45-degree turns (clockwise): N -> NE -> E -> SE -> S -> SW -> W -> NW -> N
90-degree turns (clockwise): N -> E -> S -> W -> N
4. Distance vs. Displacement
Distance (Total Path Length)
- The sum of all individual distances traveled along the path.
- Always a scalar (just a number, no direction).
- Distance >= Displacement (always).
Displacement (Shortest Distance)
- The straight-line distance from the starting point to the ending point.
- It is a vector (has both magnitude and direction).
Example
A ----3 km----> B
|
4 km
|
v
C
Distance traveled = 3 + 4 = 7 km
Displacement (AC) = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5 km
5. Pythagoras Theorem for Shortest Distance
When a person walks in two perpendicular directions, the shortest distance back to the start is found using the Pythagorean theorem:
Shortest Distance = sqrt(a^2 + b^2)
Where a and b are the perpendicular distances walked.
General Right-Angle Path
Start ---a--- Turn Point
|
b
|
End
Shortest distance (Start to End) = sqrt(a^2 + b^2)
Multiple Turns (L-shaped, U-shaped, Z-shaped paths)
For complex paths, reduce the net displacement in the North-South direction and East-West direction separately, then apply Pythagoras.
Step-by-step approach:
- Set up a coordinate system: North = +Y, East = +X, South = -Y, West = -X
- Calculate total displacement in X (East-West) direction
- Calculate total displacement in Y (North-South) direction
- Apply:
Shortest Distance = sqrt(X^2 + Y^2)
Example: Multi-Turn Path
A person walks 3 km North, then 4 km East, then 3 km South.
B -------4 km-------> C
| |
3 km 3 km
| |
A D
Net North-South displacement = 3 - 3 = 0 km
Net East-West displacement = 4 km
Shortest distance A to D = sqrt(0^2 + 4^2) = 4 km
6. Shadow-Based Direction Problems
Sunrise and Sunset Fundamentals
| Time of Day | Sun Position | Shadow Falls |
|---|---|---|
| Morning / Sunrise | East | West (opposite to sun) |
| Evening / Sunset | West | East (opposite to sun) |
| Noon (12 PM) | Directly overhead | Very short / no shadow |
Key Rule
Shadow always falls on the OPPOSITE side of the Sun.
Finding Facing Direction from Shadow
At Sunrise / Morning:
- Sun is in the East
- Shadow falls to the West
- If your shadow is in front of you -> You face West
- If your shadow is behind you -> You face East
- If your shadow is to your left -> You face North
- If your shadow is to your right -> You face South
At Sunset / Evening:
- Sun is in the West
- Shadow falls to the East
- If your shadow is in front of you -> You face East
- If your shadow is behind you -> You face West
- If your shadow is to your left -> You face South
- If your shadow is to your right -> You face North
Shadow Direction Summary Table
| Time | Sun | Shadow | Shadow in Front | Shadow Behind | Shadow Left | Shadow Right |
|---|---|---|---|---|---|---|
| Morning | East | West | Face West | Face East | Face North | Face South |
| Evening | West | East | Face East | Face West | Face South | Face North |
7. Angle-Based Direction Problems
Sometimes questions give angle of rotation instead of "left" or "right":
Angle to Direction Mapping (Clockwise from North)
| Angle (CW from N) | Direction |
|---|---|
| 0 deg | North |
| 45 deg | North-East |
| 90 deg | East |
| 135 deg | South-East |
| 180 deg | South |
| 225 deg | South-West |
| 270 deg | West |
| 315 deg | North-West |
| 360 deg | North (full circle) |
Formula
New Direction = (Current Angle + Rotation Angle) mod 360
- Clockwise rotation: Add the angle
- Counter-clockwise rotation: Subtract the angle (or add 360 - angle)
8. Direction of One Point with Respect to Another
To find the direction of point B from point A:
- Place yourself at point A
- Look toward point B
- The direction you look is "B is in ___ direction of A"
Important Distinction
- "B is to the East of A" means: standing at A, B is toward East
- "A is to the West of B" means the same thing
N
|
W ---A--- E
|
S
If B is here: B is ___ of A
B North
|
-----A-----
-----A-----
|
B South
B ---A----- West
-----A--- B East
9. Coordinate Geometry Approach
For complex problems, assign coordinates:
- Starting point = Origin (0, 0)
- North = +Y axis
- South = -Y axis
- East = +X axis
- West = -X axis
Example
Start at (0, 0)
Walk 5 km North -> (0, 5)
Walk 3 km East -> (3, 5)
Walk 2 km South -> (3, 3)
Walk 1 km West -> (2, 3)
Final position: (2, 3)
Distance from start = sqrt(2^2 + 3^2) = sqrt(4 + 9) = sqrt(13) km
Direction from start: tan(theta) = 2/3
theta = arctan(2/3) ~ 33.7 degrees from North toward East
=> North-East direction (approximately)
10. Common Pythagorean Triplets
These appear frequently in direction sense problems:
| a | b | c = sqrt(a^2+b^2) |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 6 | 8 | 10 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
| 9 | 12 | 15 |
| 9 | 40 | 41 |
| 12 | 16 | 20 |
| 15 | 20 | 25 |
Memorize at least the first 5 triplets -- they cover 90% of exam questions.
11. Special Cases and Traps
Trap 1: "Started facing North" vs. "Started walking North"
- "Facing North" means the person looks North but may not walk.
- "Walking North" means actually moving northward.
Trap 2: Left/Right vs. North/South/East/West
- Left and Right are relative to the person's facing direction.
- N/S/E/W are absolute directions.
Trap 3: "From" vs. "To"
- "Direction of A from B" = Standing at B, which direction is A?
- "Direction of A to B" = Standing at A, which direction is B?
Trap 4: Intercardinal distance
- Walking NE for
dkm does NOT adddkm to both North and East. - It adds
d/sqrt(2)to each:d * cos(45) = d * sin(45) = d / sqrt(2)