Direction Sense

Snapshot

• Direction Sense describes a person walking and turning, then asks which way they face, where they end up, or how far they are from the start. Drawn on a compass, it becomes simple geometry.
• CUET asks about five things: the direction now faced, the direction of the end point, the shortest distance, shadow and sun questions, and "A is to the north of B" landmark chains.
• The idea that cracks it: a right turn is clockwise and a left turn is anticlockwise from the way you are facing now, never from North. Draw the path with North at the top and add up the legs.
• Marking is +5 / −1, so a sketch that takes 30 seconds is cheaper than a wrong guess.

Part 1 — The compass and the turning rule

Written and reviewed by Devendra Dubey, Mathematics & Commerce faculty at UniDrill (M.Sc. Mathematics, Osmania University · 10 years teaching). Prepared for the CUET UG General Test.

Every direction question lives on one picture: North at the top of your page, East on the right, South at the bottom, West on the left, and the four diagonals (NE, SE, SW, NW) halfway between. Draw it before you read the question twice.

The only rule you must never break: a turn is measured from the direction you are facing at that moment. If you are walking South and turn left, you go East, because East is on the left-hand side of someone facing South. It is not West and it is not "left of North". The table below fixes all sixteen cases.

For turns that are not 90°, give each direction a bearing in degrees measured clockwise from North: $N = 0^\circ$, $NE = 45^\circ$, $E = 90^\circ$, $SE = 135^\circ$, $S = 180^\circ$, $SW = 225^\circ$, $W = 270^\circ$, $NW = 315^\circ$. A right turn of $\theta$ adds $\theta$, a left turn subtracts $\theta$, and you subtract or add $360^\circ$ whenever the answer leaves the range $0^\circ$ to $360^\circ$. This turns every angle question into one line of arithmetic.

Part 2 — The method, step by step

Use the same routine on every question and your speed will come from habit, not luck.

1. Draw the compass cross (a plus sign) with N at the top, and mark the start with a dot.
2. Write the facing direction at the start. If the question does not say, the first leg tells you.
3. Draw each leg as an arrow in its true direction with its length written on it.
4. Update the facing after every turn using the table or the bearing trick, and write it next to the corner.
5. Add the legs on each axis separately. East is positive and West negative on the horizontal axis. North is positive and South negative on the vertical axis.
6. Read the answer from the two net numbers: their signs give the direction (net 6 East and 8 North means North-East), and the straight-line distance is $\sqrt{h^2 + v^2}$.
7. Re-read the question to see whether it wants the direction faced, the direction of the position, the distance, or the total distance walked. These are four different answers to the same path.

Part 3 — The full menu of question types

Part 4 — Distance, Pythagoras and the triples

After step 5 of the method you hold two numbers: the net horizontal movement $h$ and the net vertical movement $v$. The straight-line distance from the start is the hypotenuse of a right triangle with those two sides.

$$d = \sqrt{h^2 + v^2}$$

The examiner almost always picks numbers that make the square root a whole number. Learn these triples by heart, along with their multiples, and you will skip the calculation.

If the legs are equal, say 12 and 12, the distance is $12\sqrt{2} \approx 17$. Distance is always positive and never larger than the total walked, because a straight line is the shortest path between two points. If the net horizontal or vertical movement is zero, the person is on a straight line from the start and no Pythagoras is needed.

Part 5 — Shadows and the sun

The sun rises in the East and sets in the West, and a shadow always falls away from the sun.

Method for shadow questions: first write which way the shadow falls (West in the morning, East in the evening). Then match that direction with the person's left, right, front or back. For example, "the shadow falls to his right in the morning" means his right-hand side is West, so he is facing South, because West is on the right of someone facing South. Whenever two people stand back to back, their facing directions are exactly opposite.

Part 6 — Worked examples

Example 1. Distance and direction. A man walks 6 km North, turns right and walks 8 km. How far is he from the start and in which direction?

North 6, then right of North is East, so East 8. Net: 8 East, 6 North, so he is North-East of the start. Distance $=\sqrt{8^2+6^2}=\sqrt{100}=$ 10 km, North-East.

Example 2. Successive 90° turns. A person faces West, turns left, then right, then right. Which direction does he face?

West, turn left gives South. South, turn right gives West. West, turn right gives North. He faces North. With bearings: $270^\circ - 90^\circ + 90^\circ + 90^\circ = 360^\circ = 0^\circ$, which is North.

Example 3. An angle turn. Facing North, a girl turns $90^\circ$ clockwise and then $135^\circ$ anticlockwise. Which direction does she face?

Bearing: $0^\circ + 90^\circ - 135^\circ = -45^\circ$, and adding $360^\circ$ gives $315^\circ$. That is North-West.

Example 4. Mixed angles. A boy faces South-East, turns $90^\circ$ anticlockwise and then $45^\circ$ clockwise. Which direction does he face?

South-East is $135^\circ$. Then $135^\circ - 90^\circ + 45^\circ = 90^\circ$, which is East.

Example 5. A three-leg path with a triple. A girl walks 20 m North, turns right and walks 5 m, then turns right again and walks 8 m. How far is she from the start?

Facing North, a right turn gives East (5 m), and another right turn gives South (8 m). Net East $=5$ m. Net North $=20-8=12$ m. Distance $=\sqrt{5^2+12^2}=$ 13 m, North-East (the 5, 12, 13 triple).

Example 6. Left, left, right. A man walks 10 km West, turns left and walks 5 km, turns left and walks 10 km, then turns right and walks 5 km. Where is he relative to the start?

West 10 puts him at 10 West of the start. Turning left from West gives South, so he walks 5 South. Turning left from South gives East, so he walks 10 East, which brings him back to the start's North-South line, 5 South of it. Turning right from East gives South, so he walks 5 more South. Final position: 10 km due South of the start, facing South. The total he walked is $10+5+10+5=30$ km, a different answer to a different question.

Example 7. Legs that cancel. From home Ravi walks 10 km East, 6 km North, 4 km West and 6 km South. Where is he now?

East-West: $+10-4=+6$ (East). North-South: $+6-6=0$. He is 6 km East of home, and no square root is needed.

Example 8. Total distance with a return trip. A runner goes 4 km North, 3 km East, and then returns to the start by the shortest route. What is the total distance run?

The return leg is the hypotenuse $\sqrt{4^2+3^2}=5$ km. Total $=4+3+5=$ 12 km.

Example 9. A landmark chain. A is to the north of B. C is to the east of B. D is to the south of C. In which direction is D from A?

Put B at the centre. A is above B. C is to the right of B. D is directly below C, so D is on the right and below the line through B. A is above that line. So D is South-East of A.

Example 10. A morning shadow. One morning, Vikram and Shailesh stand back to back on a lawn. Vikram's shadow falls exactly to his left. Which way is Shailesh facing?

Morning shadows fall West, so Vikram's left is West. A person whose left is West faces North (facing North, left is West). Shailesh stands back to back with him, so he faces South.

Example 11. An evening shadow. At sunset, Meena's shadow falls on her right-hand side. Which way is she facing?

At sunset the shadow falls East, so her right is East. Facing North puts East on the right. She faces North.

Example 12. Clock and compass. A wall clock is placed so that the 12 mark points North. In which directions do the hour and minute hands point at 4:30?

The minute hand is at 30 minutes, which is $30\times 6^\circ = 180^\circ$, so it points South. The hour hand is at $4.5\times 30^\circ = 135^\circ$, so it points South-East.

Example 13. Two walkers. Two friends start from the same point. Asha walks 15 km South and Bela walks 8 km West. How far apart are they, and where is Bela relative to Asha?

Asha is 15 below the start and Bela is 8 to its left, so the legs of the triangle are 15 and 8. Distance $=\sqrt{15^2+8^2}=$ 17 km (the 8, 15, 17 triple). Bela is above and to the left of Asha, so Bela is North-West of Asha.

Part 7 — Common traps

Part 8 — Rapid-fire practice with answers

Cover the answer column and solve each in under a minute.

Part 9 — A timed-practice routine

Direction questions reward repetition more than any other reasoning topic, because the method never changes. Follow this ladder over two weeks.

After each set, tick one cause for every wrong answer: wrong heading, wrong arithmetic, wrong question word. If the same cause appears twice, add a sentence about it to your notes. In the exam, give a direction question no more than 60 seconds. If the path has more than five legs and no triple in sight, finish the easier questions first and return.

Part 10 — Where this connects

Direction sense shares its habits with other General Test reasoning topics. The left-right discipline carries over to Ranking & Arrangement, where people face North, South or the centre of a table. The "draw it, then read it" habit is the same one used in Blood Relations. The right-triangle step and the triples are developed further in Mensuration & Basic Geometry, and the figure-rotation ideas appear again in Non-Verbal Reasoning.

One-line revision: draw with North up, rotate every turn from the current heading (right adds, left subtracts on a bearing), add each axis separately, use $\sqrt{h^2+v^2}$ for distance, and remember that shadows point away from the sun.