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.
Important Questions from Direction and Distance (Notes)
- A woman leaves her home. She walks 40 m in North-West direction and then 90 m in South-East direction. Then, she moves 30 m in North direction. How far is she now from her initial position ?
- N sits to the North of L and West of M. L sits to the East of O. If everyone is facing North, which direction of O, is N?
I starts from Point A and drives 5 km towards the north. He then takes a right turn, drives 2 km, turns right and drives 9 km. He then takes a right turn and drives 7 km. He takes a final right turn, drives 4 km and stops at Point P. How far (shortest distance) and towards which direction should he drive in order to reach Point A again? (All turns are $90^\circ$ turns only unless specified.)
A tourist drives $20 \text{ km}$ towards east, turns right and drives $6 \text{ km}$, then drives $6 \text{ km}$ towards west. He then turns to his left and drives $4 \text{ km}$ and finally turns right and drives $14 \text{ km}$. Where is he from his starting point?
- A Couple started to their offices in the morning from their home. While husband drives 3 km South and then 2 km to his right to reach his office, wife goes 2 km towards East for her office. With respect to husband's office, how far and in which direction is the wife's office ?
