Direction Sense

Prerequisites

Squares up to 25 and square roots of perfect squares — the distance step uses Pythagoras’ theorem (Class VII–X).
Addition and subtraction with signs: east/north as +, west/south as −. Basic Numeracy covers the arithmetic.
Knowing which way is east from where you are sitting right now — direction questions reward a mental map more than any formula.

The Hook: A Forest Guard’s Patrol

Connect

A forest guard (वनरक्षक) on patrol in a tiger reserve walks 7 km south from his chowki, turns left and walks 12 km, then turns left again and walks 12 km. A wireless message asks him: “How far are you from the chowki, and in which direction?”

He has walked 31 km, but that is not the answer. Turning left from south sends him east; turning left again sends him north. He is 12 km east and 5 km north of his chowki — 13 km away, towards the north-east.

The question this page answers: how do you track a person through any number of turns and give the straight-line distance and direction from the start — in under a minute?

Why it matters for CGPSC: direction sense (दिशा-ज्ञान) questions combine two quick skills — tracking turns and one line of Pythagoras. They are easy marks if you draw, and easy losses if you try to do them in your head.

The Eight Directions

On paper, north is up, east is right, south is down, west is left — always draw it this way.

North-West (उत्तर-पश्चिम)	North (उत्तर)	North-East (उत्तर-पूर्व)
West (पश्चिम)	Start	East (पूर्व)
South-West (दक्षिण-पश्चिम)	South (दक्षिण)	South-East (दक्षिण-पूर्व)

Measured clockwise from north: N = 0°, NE = 45°, E = 90°, SE = 135°, S = 180°, SW = 225°, W = 270°, NW = 315°. Opposite directions are 180° apart: NE is opposite SW, NW is opposite SE.

Turning Left and Right

A right turn is 90° clockwise; a left turn is 90° anticlockwise — both measured from the direction the person is currently facing.

Facing	After a right turn	After a left turn	After an about-turn
North	East	West	South
East	South	North	West
South	West	East	North
West	North	South	East

For turns by angles, add clockwise degrees and subtract anticlockwise degrees, then reduce to 0°–360°.

Worked example 1: turning by angles

A man faces north. He turns 90° clockwise, then 180° anticlockwise, then 45° clockwise. Which direction does he face now?

$ 0 + 90 - 180 + 45 = -45^\circ $, which is the same as $ 360 - 45 = 315^\circ $ = north-west.

Check by steps: north → east → west → north-west.

Distance and Direction from the Start

Do not trace the path step by step at the end. Add up the moves in each axis instead:

$$\boxed{x = \text{total east} - \text{total west}} \qquad \boxed{y = \text{total north} - \text{total south}}$$$$\boxed{\text{Shortest distance from start} = \sqrt{x^2 + y^2}}$$

The signs of $ x $ and $ y $ give the direction: $ x > 0, y > 0 $ → north-east; $ x < 0, y > 0 $ → north-west; and so on. If one of them is zero, the answer is a straight direction.

Exam convention for 'north-east'

In reasoning questions, “north-east” usually means “somewhere in the north-east quarter”, not exactly 45°. A point 12 km east and 5 km north is described as north-east even though the true bearing is closer to east. Only a question that asks for an angle wants more precision.

Pythagorean triples to know by sight

Sides	Distance	Multiples you will meet
3, 4	5	6–8–10, 9–12–15, 12–16–20, 15–20–25
5, 12	13	10–24–26
8, 15	17	—
7, 24	25	—
20, 21	29	—

Worked example 2

Meena walks 7 m south, turns left and walks 12 m, then turns left again and walks 12 m. How far is she from the start, in which direction, and which way is she facing?

South 7 → turn left (now east) 12 → turn left (now north) 12.

$ x = +12 $ (east), $ y = 12 - 7 = +5 $ (north). Distance $ = \sqrt{12^2 + 5^2} = \sqrt{169} = $ 13 m, towards the north-east. She is facing north after the last turn.

Worked example 3: the axis table

Facing east, Arjun walks 10 m, turns right and walks 3 m, turns right and walks 2 m, then turns left and walks 3 m. Where is he relative to the start?

Leg	Facing	East (+) / West (−)	North (+) / South (−)
1	East	+10	0
2	South	0	−3
3	West	−2	0
4	South	0	−3
Total		+8	−6

Distance $ = \sqrt{8^2 + 6^2} = 10 $ m, towards the south-east.

Shadow Questions

The sun rises in the east and sets in the west, and a shadow always falls on the side away from the sun.

Time	Sun is in the	Shadow falls towards the
Morning	East	West
Evening	West	East

Questions avoid noon, when shadows are short and their direction depends on the season and the latitude.

Method: fix the shadow’s compass direction from the time of day, then ask which way the person must face for the shadow to fall on the stated side (left, right, front or behind).

Worked example 4

One evening, Ravi’s shadow falls exactly to his right. Which direction is he facing?

Evening → shadow falls east. East is on his right, so he is facing north (facing north, your right hand points east).

Memory Tricks

Anchor these

“Never Eat Soggy Waffles” — N, E, S, W in clockwise order. A right turn moves one step along this list; a left turn moves one step back.
“Right = clockwise.” Clock hands turn the same way as a right turn.
Two axes, one root. Sum east–west, sum north–south, then one square root. Never add the legs of the walk.
3-4-5 and 5-12-13 first. If the numbers do not form one of the triples, check your arithmetic before computing an ugly root — setters usually choose triples.
“Morning shadow points west, evening shadow points east.” The shadow points away from the sun.

Don’t Confuse

This	vs	That
Distance travelled — sum of all legs		Distance from start — straight line $ \sqrt{x^2 + y^2} $
Turn “left” relative to the person’s facing		“West”, a fixed compass direction
Final position direction (where he is)		Final facing direction (where he looks)
North-west = 315°		North-east = 45°
Morning shadow → west		Evening shadow → east

Common Mistakes

Trap 1: Turning from the wrong direction

“Turns right” means right of the way the person is facing now, not right of north. A person walking south who turns right goes west. Write the facing direction on every leg of your sketch.

Trap 2: Mixing position and facing

“In which direction is he from the start?” asks about his position; “Which direction is he facing?” asks about his last leg. In Worked example 2, Meena is north-east of the start but facing north. Options include both.

Trap 3: Reversing the reference point

“Where is A from B?” puts B at the centre. If A is north-east of B, then B is south-west of A. Read which point is the reference before answering.

Practice

Level 1: Foundation

A man faces east and turns right. Which direction does he face now?
Which direction is opposite to south-east?
A person walks 3 km north and then 4 km west. How far is he from the start?
Facing north, a girl turns 135° clockwise. Which direction does she face now?
In the morning, a pole’s shadow falls in which direction?

Answers

South.
North-west.
5 km (3-4-5 triple), towards the north-west.
South-east (135°).
West.

Level 2: Prelims MCQs

Q1. Starting from his house, Rohit walks 8 m east, turns left and walks 9 m, turns left and walks 4 m, turns left and walks 6 m, and finally turns right and walks 8 m. How far and in which direction is he from his house?

(a) 5 m north-west (b) 5 m north-east (c) 7 m north-west (d) 3 m west

Answer: (a) 5 m north-west

Facing: east → north → west → south → west.

East–west: $ +8 - 4 - 8 = -4 $ (4 m west). North–south: $ +9 - 6 = +3 $ (3 m north). Distance $ = \sqrt{4^2 + 3^2} = 5 $ m, towards the north-west.

Q2. One morning, Ravi and Sohan were standing face to face in a field. Ravi’s shadow fell exactly to his left. In which direction was Sohan facing?

(a) North (b) South (c) East (d) West

Answer: (b) South

Morning → shadows fall west. West is on Ravi’s left, so Ravi faces north. Sohan faces Ravi, so Sohan faces south. Option (a) is Ravi’s direction — the “answered for the wrong person” trap.

Q3. From point P, a cyclist rides 4 km east to Q, turns right and rides 3 km to R, then turns right and rides 8 km to S. Consider the following statements:

S is 5 km from P.
S is to the south-west of P.
The distance between Q and S is 8 km.

Which of the statements given above is/are correct?

(a) 1 only (b) 2 and 3 only (c) 1 and 3 only (d) 1 and 2 only

Answer: (d) 1 and 2 only

Take P as the origin. Q = (4, 0); turning right from east means south, so R = (4, −3); turning right from south means west, so S = (−4, −3).

P to S: $ \sqrt{4^2 + 3^2} = 5 $ km (statement 1 correct), and S is west and south of P → south-west (statement 2 correct). Q to S: east–west gap 8 km, north–south gap 3 km, so $ \sqrt{64 + 9} = \sqrt{73} \approx 8.5 $ km, not 8 (statement 3 wrong).

Q4. A girl is facing south-east. She turns 135° clockwise and then 270° anticlockwise. Which direction is she facing now?

(a) South (b) West (c) North (d) North-east

Answer: (c) North

South-east = 135°. $ 135 + 135 - 270 = 0^\circ $ = north. (After the first turn she faces west, 270°; turning 270° anticlockwise from west brings her to north.)

Q5. A person walks the two perpendicular legs in List I. Match each with the straight-line distance from the start in List II:

List I (legs)	List II (distance)
A. 6 km and 8 km	1. 13 km
B. 5 km and 12 km	2. 25 km
C. 8 km and 15 km	3. 10 km
D. 7 km and 24 km	4. 17 km

(a) A-3, B-4, C-1, D-2 (b) A-1, B-3, C-4, D-2 (c) A-3, B-1, C-4, D-2 (d) A-3, B-1, C-2, D-4

Answer: (c) A-3, B-1, C-4, D-2

These are the standard triples: 6–8–10, 5–12–13, 8–15–17 and 7–24–25. Recognising them saves the square-root step entirely.

Level 3: Speed Drill — Solve in 60 Seconds

Each item should take under 60 seconds with a rough sketch. In Mains Paper IV (GS-II), the sketch with the axis totals is your working.

Facing west, a man turns left. Which way does he face?
Walk 6 km east, then 8 km north. Distance and direction from the start?
Facing north-east, turn 90° anticlockwise. Which way do you face?
Walk 5 m north, make an about-turn and walk 12 m. Where are you from the start?
In the evening, your shadow falls directly behind you. Which way are you facing?
Walk 12 m south, then 5 m west. Distance and direction from the start?