One day, a man left home for a morning walk and went 12 km south. He then…

Concept

In direction problems, represent each movement by horizontal and vertical components. Movements in opposite directions on the same axis cancel by their signed distances.

The straight-line distance from the final point to the start is the magnitude of the net displacement, not the total path length.

Application

1. Take the home as (0, 0). Moving 12 km south gives (0, -12).

2. While facing south, a right turn points west. Moving 8 km gives (-8, -12).

3. While facing west, a right turn points north. Moving 12 km gives (-8, 0).

4. While facing north, a left turn points west. Moving 10 km gives (-18, 0).

5. The vertical displacement is 0 km and the horizontal displacement is 18 km west, so the direct distance to home is 18 km.

Cross-check

From (-18, 0), walking 18 km east returns to (0, 0), which confirms the displacement.

Therefore, the man must walk 18 km in a straight line to reach home.