To determine the shortest distance and direction from Point P back to Point A, we need to track Arvind's movements using a coordinate system.

Mapping Arvind's Driving Path

Let's assume Point A is the origin (0, 0) on a 2D plane, with East along the positive x-axis and North along the positive y-axis.

• Start at A: (0, 0)
• Move 1: 25 km East. New position: (0 + 25, 0) = (25, 0).
• Move 2: Takes a right turn (now facing South), drives 10 km. New position: (25, 0 - 10) = (25, -10).
• Move 3: Turns left (now facing East), drives 20 km. New position: (25 + 20, -10) = (45, -10).
• Move 4: Turns left (now facing North), drives 10 km. New position: (45, -10 + 10) = (45, 0).
• Move 5: Takes a final left turn (now facing West), drives 12 km. New position: (45 - 12, 0) = (33, 0). This is Point P.

Calculating Return Journey from P to A

Arvind's final position is Point P at coordinates (33, 0). Point A is at the origin (0, 0).

To return from Point P (33, 0) to Point A (0, 0), Arvind needs to cover the displacement from P to A.

• Change in East-West direction (x-axis): $0 - 33 = -33$ km. A negative value indicates movement towards the West.
• Change in North-South direction (y-axis): $0 - 0 = 0$ km.

The required movement is 33 km towards the West.

Determining Shortest Distance and Direction

The shortest distance is the magnitude of the displacement vector ($-33, 0$).

Distance = $\sqrt{(-33)^2 + (0)^2} = \sqrt{1089} = 33$ km.

The direction is determined by the displacement vector components. Since the x-component is -33 (West) and the y-component is 0, the direction is West.

Therefore, Arvind needs to drive 33 km towards West to reach Point A again.