Sunita starts from a fixed point and travels 6 km towards the north-east, then 9 km towards the west, then 6 km towards the south-west and finally 3 km towards the east. How far is she from her initial position?

Introduction / Context:
In this direction sense question, Sunita walks in four different directions, including two diagonal directions. The task is to calculate how far she is from her initial starting point after completing the full path. This type of problem tests spatial visualization, vector addition and understanding of north-east and south-west movements on a coordinate plane.

Given Data / Assumptions:
- Sunita starts from an origin point, which we can treat as (0, 0) on a coordinate grid. - First she travels 6 km towards the north-east. - Then she travels 9 km towards the west. - Next she travels 6 km towards the south-west. - Finally she travels 3 km towards the east. - Directions follow the usual convention: east is positive x, north is positive y.

Concept / Approach:
To solve this, we break each movement into horizontal (east west) and vertical (north south) components. Movements purely west or east affect only the x coordinate. Movements purely north or south affect only the y coordinate. Movements towards north east or south west are along a 45 degree diagonal, so their x and y components have equal magnitude. By adding all horizontal components together and all vertical components together, we obtain the final coordinates. The straight line distance from the starting point is then the length of the resulting vector. In this particular case the vertical components cancel and the horizontal components simplify to an exact integer distance.

Step-by-Step Solution:
1. Represent the north east movement of 6 km as equal components: 6 / sqrt(2) km east and 6 / sqrt(2) km north. 2. After this first move Sunita is at approximately (4.24 km east, 4.24 km north). 3. The second move is 9 km west, which reduces only the x coordinate: new x = 4.24 - 9 ≈ -4.76 km, y remains about 4.24 km. 4. The third move is 6 km towards the south west, again at 45 degrees, so the components are 6 / sqrt(2) km west and 6 / sqrt(2) km south. 5. Subtract these components: x ≈ -4.76 - 4.24 = -9 km, y ≈ 4.24 - 4.24 = 0 km. 6. The fourth move is 3 km east, so x = -9 + 3 = -6 km, y remains 0 km. 7. The final position is therefore 6 km to the west of the starting point on the same horizontal line.

Verification / Alternative check:
We can avoid decimals by reasoning symbolically. The first and third diagonal moves are equal in length and opposite in direction: north east versus south west. Their vertical components cancel exactly and their horizontal components also cancel. Therefore, these two diagonal legs together contribute net zero displacement. What remains is simply a 9 km west movement followed by a 3 km east movement. That simplifies to a net 6 km west displacement. A quick sketch on squared paper will show that Sunita ends up directly west of the start, on the same east west line, confirming the reasoning.

Why Other Options Are Wrong:
- 3 km is half of the correct horizontal displacement and would arise from incorrectly subtracting 9 - 6 instead of 9 - 3. - 9 km would be correct only if the final 3 km east movement were ignored. - 12 km assumes that all west movements are simply added without considering the final east movement. - 15 km is an overestimate that ignores both cancellation of diagonal legs and the final balancing movement.

Common Pitfalls:
A common mistake is to treat diagonal movements as if they were purely horizontal or purely vertical, which changes the total displacement. Another frequent error is to sum all distances walked instead of the net displacement, forgetting that the question asks for the straight line distance from the starting point, not total distance travelled. Some learners also forget that opposite legs of equal length in opposite diagonal directions cancel out when computing net displacement. Careful separation of x and y components avoids all these errors.

Final Answer:
Sunita is 6 km away from her initial position, directly to the west of the starting point.