A and B start running from the same point. A runs 3 km towards the west, then turns towards the south and runs 5 km, and then turns to her right and runs 7 km. B runs 1 km towards the south, then turns to her right and runs 10 km. After all these movements, where is B located with respect to A in terms of direction and distance?

Introduction / Context:
This direction sense problem involves two people, A and B, who run different paths starting from the same point. We are not asked for their positions relative to the starting point, but specifically for the position of B with respect to A. This requires computing both final coordinates and then subtracting to get the relative displacement vector from A to B.

Given Data / Assumptions:

• A runs 3 km towards the west from the starting point.
• A then turns south and runs 5 km.
• A then turns to her right and runs 7 km.
• B runs 1 km towards the south from the starting point.
• B then turns to her right and runs 10 km.
• Right turns are defined relative to the current facing direction of the runner.
• All movements are on straight, level ground.

Concept / Approach:
We assign a coordinate system with the starting point at (0, 0). Using this, we calculate the final coordinates of A and B separately. A westward movement decreases the horizontal coordinate, while eastward movement increases it. Similarly, north and south movements affect the vertical coordinate positively or negatively. Once we have the coordinates of A and B, we find the displacement from A to B by subtracting A position from B position and interpret the resulting vector as direction and distance.

Step-by-Step Solution:
Step 1: Track A. From (0, 0) A runs 3 km west to (−3, 0). Step 2: A turns south and runs 5 km, reaching (−3, −5). Step 3: A then turns right. Facing south, a right turn makes her face west. She runs 7 km west, reaching (−3 − 7, −5) = (−10, −5). Step 4: Now track B. From (0, 0) B runs 1 km south to (0, −1). Step 5: B then turns right. Facing south, a right turn means B faces west. B runs 10 km west to reach (−10, −1). Step 6: A final position is (−10, −5). B final position is (−10, −1). From A to B, the horizontal coordinate is the same and the vertical changes from −5 to −1, which is a movement of 4 km towards the north.

Verification / Alternative check:
Notice that both A and B end up on the same vertical line: each is 10 km west of the starting point. The only difference is their north south coordinate. A is 5 km south of the origin and B is 1 km south of the origin. Thus B is 4 km north of A. This matches the coordinate calculation and provides a simple visual interpretation if you sketch the points on paper.

Why Other Options Are Wrong:
The options mentioning 4 km south or 6 km south reverse the actual relative direction. The distances of 6 km north or south arise from miscalculating one of the vertical displacements, such as mistaking A southward run length or mixing up B path. Only 4 km north correctly reflects the difference between the two final vertical positions.

Common Pitfalls:
A frequent error is to compute each runner position relative to the origin but then forget to take their difference. Some learners also misapply right turns or confuse the path segments. Always carefully write down A and B coordinates and then compute B minus A to get the relative position.

Final Answer:
B is located 4 km north of A.