Important Formula Direction Sense - General Intelligence and Reasoning

Formulas For Directional Senses

Directional sense is the ability to understand and reason about positions and movements relative to the four cardinal directions - North, South, East, and West - and the four intercardinal directions - North-East, South-East, South-West, and North-West. This skill is frequently tested in reasoning and aptitude sections of competitive examinations and is essential for solving problems about turns, relative position, distance and resultant displacement.

Definition: The four primary directions are North, South, East and West. The positions halfway between each pair of primary directions are called intercardinal directions: North-East, South-East, South-West and North-West.

Important facts about directional sense

For problem solving, assume a standard orientation: front = North, back = South, left = West, right = East, unless the problem states a different initial facing.

• Sun and shadow rule: the sun rises in the East and sets in the West. In the morning the sun is roughly towards the East, so shadows fall towards the West. In the evening shadows fall towards the East.
• Turning angles: a turn to the immediate left or right from any main direction is a 90° change. Turning from a main direction to an adjacent intercardinal direction (for example North to North-East) is a 45° change.
• Relative facing: when two people are face to face, they face opposite directions; when they are back to back, they face opposite directions but away from each other. The meaning of left and right is always relative to the person's current facing direction.
• When problems describe distances moved in perpendicular directions (for example, some distance north and some distance east), the straight-line distance between start and end can be found using the Pythagoras theorem.
• Assume all movements are along straight lines and follow the stated turns unless the problem says otherwise.

Concept: Pythagoras theorem 

The Pythagoras theorem is a fundamental result of Euclidean geometry that applies to right-angled triangles. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

The relation in mathematical  form is: \(c^{2} = a^{2} + b^{2}\).

In directional problems, if a person moves a units in the north-south direction and b units in the east-west direction (and those displacements are perpendicular), then the straight-line distance from the starting point to the final position is given by \( \sqrt{a^{2} + b^{2}} \). This is useful when a result of movements forms a right triangle (for example, when net north and net east displacements are known).

Example application: if a person walks 20 m north and then 15 m east, the straight-line distance between start and end is \( \sqrt{20^{2} + 15^{2}} = \sqrt{400 + 225} = \sqrt{625} = 25\ \text{m} \).

Solved Examples:

Q1: One morning Udai and Vishal were talking to each other face to face at a crossing. If Vishal's shadow was exactly to the left of Udai, which direction was Udai facing?
(a) East
(b) West
(c) North
(d) South

Ans: (c)

Sol.
In the morning the sun is towards the East, so shadows fall towards the West.
Vishal's shadow is therefore to the West of Vishal.
Udai and Vishal are face to face, so they face opposite directions.
If the point to Vishal's west is to Udai's left, then Udai's left must be West.
If Udai's left is West, then Udai is facing North.
Therefore Udai was facing North.

Q2: If Rajesh is  facing west, and turn 90 degrees to his right which direction will be he facing?
(a) East
(b) North
(c) South
(d) West

Ans: (b)

Sol.
Rajesh is initially facing West.
A 90° turn to the right from West faces North.
Therefore he will be facing North.

Q3: If you're standing at point A which is in South and you want to go to point B, which direction should you move if point B is to the right of point A?
(a) East
(b) West
(c) North
(d) south

Ans: (a)

Sol.
Assume the standard map orientation where North is up and East is to the right.
If point B is to the right of point A on such a map, then B is towards the East of A.
Therefore you should move East to go from A to B.

Q4: Ravi went to a park, and walked 25km towards North, then he turned right and walked 30 m , then he turned right and walked 35 m the  he turned left and and walked 15 km and finally turns left and walks 15 km. In which direction and how many meters is he  from the starting position?
(a) 15 m West
(b) 30 m East
(c) 30 m West
(d) 45 m East

Ans: (d)

Sol.
The question mixes units (km and m). The answer choices use metres, so to compare with the options we interpret the distances as metres (this interpretation matches the given answer).
Represent the moves step by step and compute net east-west and north-south displacements.
Start at (0,0).
1) Walk 25 (assumed m) towards North → position (0,25).
2) Turn right (from North → right is East) and walk 30 m → position (30,25).
3) Turn right (from East → right is South) and walk 35 m → position (30,25-35) = (30,-10).
4) Turn left (from South → left is East) and walk 15 (assumed m) → position (30+15, -10) = (45,-10).
5) Turn left (from East → left is North) and walk 15 (assumed m) → position (45, -10+15) = (45,5).
Net east displacement = 45 m; net north displacement = 5 m.
Straight-line distance from start = \( \sqrt{45^{2} + 5^{2}} = \sqrt{2025 + 25} = \sqrt{2050} \approx 45.28\ \text{m} \).
The resultant direction is slightly north of east (about 6.34° north of east), but the closest option among those given is 45 m East.
Hence option (d) is the best match.

Q5: Walking 15m North, Rahul turned towards right and walked for 30 m , then again turned left and walked 45 m and finally turned right . In which direction is he moving from his starting position?
(a) West
(b) East
(c) North
(d) South

Ans: (b)

Sol.
Track Rahul's facing direction step by step.
1) After walking 15 m North, he is facing North.
2) He turns right; from North, right is East, and walks 30 m facing East.
3) He then turns left; from East, left is North, and walks 45 m facing North.
4) Finally he turns right; from North, right is East.
Therefore, after the final turn he is facing (and hence moving or ready to move) towards East.

Practical shortcuts and tips

• Always draw a simple sketch; mark initial facing and every turn (left/right) relative to the current facing.
• Use vectors: keep track of net east-west (x) and north-south (y) displacements separately. East and North as positive; West and South as negative. Final distance = \( \sqrt{x^{2} + y^{2}} \).
• Remember the sun-shadow rule for morning/evening shadow questions: morning → sun in East → shadows to West; evening → sun in West → shadows to East.
• When units are mixed or unclear in a question but options use a single unit, convert appropriately or assume consistent units as indicated by options (state the assumption in your working).
• For multiple turns, track the current facing direction rather than always referring to the original facing.

Summary

Direction sense problems require clear tracking of facing, correct application of left/right rules, use of the sun-shadow rule where relevant, and application of the Pythagoras theorem for perpendicular displacements. Practise sketching and vector bookkeeping to solve these problems quickly and accurately.