A person walks 100 m straight from his house, turns left and walks 100 m, again turns left and walks 300 m, then turns right and walks 100 m to reach his office. In which direction does he walk initially from his house if his office is exactly in the North-East direction?
- (a) North-West
- (b) West
- (c) South ✓ UPSC's answer
- (d) South-West
Why the answer is (c)
• Let the initial direction be $D$. The path consists of four segments: 100m in $D$, 100m in $D+90^\circ$ (left), 300m in $D+180^\circ$ (left again), and 100m in $D+270^\circ$ (right from $D+180^\circ$).
• Assigning coordinates with North as +Y and East as +X, let the initial direction be an angle $\theta$ from the East axis.
• The net displacement vector is the sum of the four vectors. For the office to be in the North-East direction, the net displacement must have equal positive X and Y components (slope of 1, i.e., $45^\circ$ from East).
• Testing option (c) South: Initial vector is $(0, -100)$. Left turn (East): $(100, 0)$. Left turn (North): $(0, 300)$. Right turn (East): $(100, 0)$. Net displacement: $(100+100+0+0, -100+0+300+0) = (200, 200)$. This is exactly North-East.
• Therefore, the initial direction must be South.
Why the other options are wrong
- (a) North-West
- If the initial direction is North-West, the net displacement does not result in a North-East direction relative to the start.
- (b) West
- If the initial direction is West, the net displacement vector points towards the North-West, not North-East.
- (d) South-West
- If the initial direction is South-West, the net displacement vector does not align with the North-East quadrant.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2024, held on 16 June 2024. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.