Starting from a hotel, Bhawesh proceeded in a straight path and then made a left turn after...
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Starting from a hotel, Bhawesh proceeded in a straight path and then made a left turn after covering 3 km. He continued for 2 km in that direction, arriving at Point X.
At Point X, he turned right and traveled 4 km, followed by another right turn and 4 km of travel. From this location, he moved directly towards Point X.
If Bhawesh's movement at that stage is towards the North-West direction, what was the initial direction in which he set out from the hotel?
Show answer & explanation
The final movement from the last point back to Point X is North-West. By tracing the movements backward from Point X, we determine that Point X must be North-West of the final position, which implies the orientation of the path. This aligns with an initial starting direction of East.
Step-by-step Derivation:
Step 1: Define the coordinates of Point X as (0, 0). Let the initial direction be 'D'.
Step 2: Trace the movement from Point X:
- From X(0,0), he turns right and travels 4 km. Let's assume the direction he was facing at X was 'F'. A right turn from 'F' leads to a direction 'F+90°'.
- He then turns right again and travels 4 km. This direction is 'F+180°'.
- Let the final position be P. P = (0,0) + 4(unit vector of F+90°) + 4(unit vector of F+180°).
Step 3: The problem states he moves from P directly back to X(0,0) in the North-West direction. Therefore, the vector PX is North-West. This means the vector XP (from X to P) is South-East.
Step 4: If XP is South-East, and XP = 4(unit vector of F+90°) + 4(unit vector of F+180°), we can deduce F. If F is North, then F+90° is East and F+180° is South. The vector (4 East + 4 South) results in a South-East direction. Thus, Bhawesh was facing North when he arrived at Point X.
Step 5: Trace back to the hotel. To arrive at Point X facing North, he must have been traveling North for the last 2 km. Before that, he made a left turn to face North. A left turn to face North means he was previously traveling East.
Step 6: Since he traveled in a straight path from the hotel before that left turn, his initial direction from the hotel was East.