A plane first flies 100 km towards the west from its starting point, then turns towards the south and flies 150 km, then turns again towards the west and flies another 300 km, and finally turns to its right and flies 150 km. After completing all these legs of the journey, where is the plane now with reference to its starting position, in terms of direction and distance?

Introduction / Context:
This is another displacement and direction sense question, but with larger distances that resemble the route of an aircraft. The goal is to find the final position of the plane relative to the starting point after flying several legs in different directions. Only by analysing the net horizontal and vertical movements can we identify both the correct distance and the correct direction.

Given Data / Assumptions:

• The plane flies 100 km towards the west from its starting point.
• It then turns south and flies 150 km.
• Next, it turns west again and flies 300 km.
• Finally, it turns to its right and flies 150 km.
• Right is defined relative to its current flight direction at that moment.
• The flight is assumed to be in a horizontal plane relative to the map directions without changes in altitude affecting the calculation.

Concept / Approach:
We model the starting point as the origin in a coordinate system. Movements west decrease the horizontal coordinate, east increases it, south decreases the vertical coordinate, and north increases it. Right turns are always made relative to the aircraft current heading. When it has moved equal distances south and north, those effects cancel out, leaving a purely east west displacement. Our task is to compute that net displacement.

Step-by-Step Solution:
Step 1: Start at (0, 0). Flying 100 km west brings the plane to (−100, 0). Step 2: The plane then turns south and flies 150 km, reaching (−100, −150). Step 3: Next, it turns west again and flies 300 km, so the horizontal coordinate decreases by another 300. The new position is (−100 − 300, −150) = (−400, −150). Step 4: At this point, the plane is facing west. A right turn from west means it now faces north. It then flies 150 km north, which increases the vertical coordinate from −150 to 0, so the final position becomes (−400, 0). Step 5: The starting point was (0, 0). Therefore, the plane is now 400 km west of the starting point with no net north south displacement.

Verification / Alternative check:
Consider the north south movements. The plane goes 150 km south and later 150 km north, which completely cancel out. In the east west direction, it travels 100 km west and then an additional 300 km west, with no eastward movement at all. Thus the total westward displacement is 100 + 300 = 400 km. This directly confirms that the final position is 400 km west of the origin.

Why Other Options Are Wrong:
The options 400 km east and 200 km east reverse the actual direction, which is west. The options 200 km west and 200 km east reflect partial calculations where only part of the westward journey is considered or the cancellation of north and south movements is mishandled. Therefore, none of these represent the full net displacement correctly except 400 km west.

Common Pitfalls:
Common errors include ignoring that equal north and south movements negate each other, and mistakenly subtracting only part of the westward legs. Some test takers also confuse the right turn from west, assuming it leads to south instead of north. Drawing a simple diagram of the path helps maintain clarity.

Final Answer:
The plane is now located 400 km west of its starting position.