Pavan starts from Point A and drives 10 km towards North. He then takes a left…

In direction-sense problems, model every straight segment of the journey as a displacement vector on a compass-aligned coordinate plane, with North taken as the positive vertical axis and East as the positive horizontal axis. A left turn always rotates the current heading 90 degrees anticlockwise (North to West to South to East to North), and a right turn rotates it 90 degrees clockwise. Adding up all the vector displacements gives the traveller's final position, and the straight-line distance plus the compass direction needed to return to the starting point follow directly from the net vertical and horizontal offsets between the final position and the start.

1. Place Point A at the origin (0, 0). Facing North, Pavan drives 10 km, moving to (0, 10).

2. Facing North, a left turn changes his heading to West; driving 6 km West moves him to (-6, 10).

3. Facing West, a left turn changes his heading to South; driving 12 km South moves him to (-6, -2).

4. Facing South, a left turn changes his heading to East; driving 9 km East moves him to (3, -2).

5. Facing East, a final left turn changes his heading to North; driving 2 km North moves him to (3, 0), which is Point P.

6. Comparing P (3, 0) with A (0, 0) shows a vertical offset of 0 km and a horizontal offset of 3 km, with P positioned to the east of A.

As an independent check, sum the North-South legs separately from the East-West legs. North-South: +10 km (North) - 12 km (South) + 2 km (North) = 0 km, so P and A share the same east-west line. East-West: -6 km (West) + 9 km (East) = +3 km, so P lies 3 km east of A. Both methods agree that P is displaced 3 km east of A with no north-south offset.

Since P lies 3 km to the east of A on the same horizontal line, Pavan must drive 3 km to the west to return to Point A.