Vikas starts from Point A and drives 11 km towards north. He then takes a right turn, drives 13 km, turns right and drives 12 km. He then takes a right turn and drives 17 km. He takes a final right turn, drives 1 km and stops at Point P. How far (shortest distance) and towards which direction should he drive in order to reach Point A again? (All turns are 90-degree turns only unless specified.)
4 km to the east
Place Point A at the origin (0, 0) with East as +x and North as +y. Driving 11 km north takes Vikas to (0, 11).
A right turn while facing north makes him face east; 13 km east takes him to (13, 11). A right turn while facing east makes him face south; 12 km south takes him to (13, 11 – 12) = (13, –1).
A right turn while facing south makes him face west; 17 km west takes him to (13 – 17, –1) = (–4, –1). The final right turn while facing west makes him face north; 1 km north takes him to Point P = (–4, 0).
Point P = (–4, 0) lies exactly 4 km west of Point A = (0, 0). So to return to A he must drive 4 km towards the east.
Check: net north–south movement = 11 – 12 + 1 = 0 km and net east–west movement = 13 – 17 = –4 km (4 km west), so the shortest path back is 4 km in the opposite direction, east.
Hence, 4 km to the east.
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