A man walks 5 km towards the South and then turns to his right. After walking 3 km, he turns left and walks 4 km. He then goes back 10 km straight along the same line. From his starting place, in which direction is he now located?

Introduction:
This direction–sense problem describes a person walking in several legs, with some left and right turns, and finally retracing a path in the opposite direction. We must determine the man's final direction from the starting point. The key idea is to treat each segment as a vector and keep track of both direction and distance using a coordinate system.

Given Data / Assumptions:
• First, the man walks 5 km towards the South.• He then turns to the right and walks 3 km.• Next, he turns left and walks 4 km.• Finally, he goes back 10 km straight, meaning directly opposite to his then current facing direction.• We assume standard compass directions and level ground.

Concept / Approach:
We place the starting point at the origin of an x–y coordinate system: North as positive y, East as positive x, South as negative y, and West as negative x. By updating his position after each move and turn, we can find his final coordinates. The direction from the origin to this final point tells us in which quadrant he is (for example, North-East, North-West), which is what the question asks for.

Step-by-Step Solution:
Step 1: Start at (0, 0) facing South. Walking 5 km South takes him to (0, −5).Step 2: Turning right from South makes him face West. Walking 3 km West moves him to (−3, −5).Step 3: Turning left from West makes him face South again. Walking 4 km South takes him to (−3, −9).Step 4: He then "goes back 10 km straight". At this point he is facing South, so going back means turning around and walking 10 km towards the North.Step 5: From (−3, −9), moving 10 km North increases the y-coordinate: −9 + 10 = 1. His final position is (−3, 1).Step 6: Relative to the origin, he is 3 km West (negative x) and 1 km North (positive y). This lies in the North-West quadrant.

Verification / Alternative check:
We can summarize the net movement: along the north–south axis he moved 5 km South, then 4 km further South, and finally 10 km North. The combined effect is 9 km South and 10 km North, giving a net 1 km North. Along the east–west axis he moved only 3 km West, with no eastward movement, giving a net 3 km West. The combination of North and West components confirms the direction as North-West.

Why Other Options Are Wrong:
South-East would require net movement South and East, which is inconsistent with a final y-coordinate of +1 and x-coordinate of −3. Pure South would need no net northward movement, but we have a net movement of 1 km North. West alone would require no net northward displacement. North-East would involve positive x and positive y, whereas we have negative x and positive y. Only North-West matches the coordinate signs.

Common Pitfalls:
Many learners forget that "goes back 10 km straight" means reversing direction along the same line, not continuing forward. Others lose track of left and right turns, especially when the person is facing South or West. Drawing a labelled sketch with each leg marked is a simple and effective way to avoid such directional mistakes.

Final Answer:
From the starting place, the man is finally located in the North-West direction.