Raj is facing north. He cycles 8 km towards the north, then moves 10 km in the south-east direction and finally 6 km towards the west. In which direction is he from his original position now?

Introduction / Context:
This problem combines straight northward motion, a diagonal motion towards the south east, and a final straight westward motion. We are asked only for Raj's final direction relative to his starting point, not the exact distance. The diagonal component makes the motion two dimensional and requires careful handling of both horizontal and vertical movements.

Given Data / Assumptions:
- Raj starts from an original position facing north. - He cycles 8 km towards the north. - He then moves 10 km towards the south east. - Finally he moves 6 km towards the west. - South east is a diagonal direction halfway between south and east, making a 45 degree angle with each axis.

Concept / Approach:
We use a coordinate system where east is positive x and north is positive y. Northward and westward segments are easy to represent, while the south east movement must be broken into equal horizontal and vertical components because it is at 45 degrees. By summing all x components and all y components, we obtain the final coordinate. The signs of the resulting x and y tell us the final direction: positive x and positive y mean north east, and so on.

Step-by-Step Solution:
1. Start at (0, 0). 2. Move 8 km north: new position is (0, 8). 3. Move 10 km south east. At 45 degrees, the components are 10 / sqrt(2) km east and 10 / sqrt(2) km south. 4. Numerically, 10 / sqrt(2) is about 7.07 km. Adding components, x becomes 0 + 7.07 = 7.07 and y becomes 8 - 7.07 ≈ 0.93. 5. Move 6 km west: subtract 6 from the x coordinate. New x = 7.07 - 6 = 1.07, y remains about 0.93. 6. Final position is approximately (1.07, 0.93). Both coordinates are positive, so Raj is to the east and north of the starting point. 7. Since both x and y are positive, his final direction from the origin is north east.

Verification / Alternative check:
Even without exact decimals, we know that the south east move reduces some of the northward distance and introduces eastward displacement. After 8 km north, moving approximately 7.07 km south reduces the net north movement to around 0.93 km. The same diagonal leg adds about 7.07 km east. Subtracting the 6 km west leaves about 1.07 km east. Having a small net north and a small net east displacement shows that Raj is in the north east quadrant relative to the start. A rough scaled diagram of the three legs confirms that the endpoint lies slightly north and slightly east of the origin.

Why Other Options Are Wrong:
- South East would require the final y coordinate to be negative, which it is not. - North West would require the x coordinate to be negative, which contradicts the final positive x value. - South West would require both x and y to be negative, which is not the case. - At the original position would mean both net x and net y are zero, which they clearly are not after unequal movements.

Common Pitfalls:
A frequent error is treating the 10 km south east leg as 10 km purely south or purely east instead of splitting it into components. Another mistake is to subtract 6 km west directly from 10 km south east without considering that the diagonal leg already included an east component. Carefully separating movements into x and y components and checking signs ensures correct classification of the final direction.

Final Answer:
Raj is to the north east of his original position.