Amar started from point A and walked 10 km East to point B then turned to…

A directional (direction-test) problem is solved by resolving each leg of the path into two independent components -- one along the East-West axis and one along the North-South axis -- and summing each axis separately. The endpoint's direction from the start is fixed purely by the signs and magnitudes of these two net components, not by the order of the turns.

1. Amar walks 10 km East from A to B: horizontal displacement = 10 km East.

2. He turns North and walks 3 km to C: vertical displacement = 3 km North.

3. He turns West and walks 12 km to D: the horizontal running total becomes 10 km East minus 12 km West = 2 km West.

4. He turns South and walks 3 km to E: the vertical running total becomes 3 km North minus 3 km South = 0 km.

Cross-check with coordinates: taking A as the origin (0, 0) with East as +x and North as +y -- B = (10, 0), C = (10, 3), D = (10 - 12, 3) = (-2, 3), E = (-2, 3 - 3) = (-2, 0). E relative to A is (-2, 0), i.e. 2 km along the negative x-axis, matching the component method above.

Since the net vertical displacement is zero and the net horizontal displacement is 2 km along the negative x-axis, point E lies directly to the west of the starting point A.