A person facing the East travels 4 km straight and then turns right and travels 3 km, then further turns left and travels 2 km and finally turns left and travels 3 km. The minimum distance between the final point and the initial point, and the direction in which the person is facing at the final point are, respectively

The correct option is D - 6 km, North.

[as per provisional answerkey]

Step-by-step Trace

Let the starting point be A (0, 0) on a Cartesian plane, where East is the positive x-axis and North is the positive y-axis.

• Step 1: Starting at A, the person faces East and travels 4 km.
  New position: B (4, 0).
• Step 2: Turns right (which is South) and travels 3 km.
  New position: C (4, -3).
• Step 3: Turns left (while facing South, a left turn points East) and travels 2 km.
  New position: D (6, -3).
• Step 4: Finally, turns left (while facing East, a left turn points North) and travels 3 km.
  Final position: E (6, 0).

Final Direction: In the last step, the person turned left from East and traveled 3 km. Therefore, the person is facing North.

Minimum Distance: The minimum distance is the straight line between the initial point A (0, 0) and the final point E (6, 0).
Distance = 6 - 0 = 6 km.

Why the other options are incorrect

• Option (a) - 12 km, East: This incorrectly calculates the total path distance ($4+3+2+3=12$) instead of the displacement (minimum distance) and misidentifies the final facing direction.
• Option (b) - 6 km, East: While the distance is correct, it fails to recognize that the final 3 km movement was towards the North after a left turn from an Eastward heading.
• Option (c) - 8 km, North: This is a calculation error in the horizontal displacement; the person moved 4 km and then another 2 km in the Eastward direction, totaling 6 km, not 8 km.

Key Concept

Displacement vs. Distance: Minimum distance refers to the straight-line displacement between two points, and direction is determined by the final orientation after the last turn.