Rakesh left home and walked $5$km southwards, then turned right and walked $2$km and again turned right and walked $5$ km and finally again turned left and walked $5$ km. The shortest distance between the final position and home is.
Understanding Rakesh's Directional Journey
Rakesh's movement can be broken down into distinct steps. We need to track his position relative to his starting point (home) to find the shortest distance.
- Step 1: Rakesh starts at home and walks 5 km South.
- Step 2: He turns right from the South direction (which means he faces West) and walks 2 km West.
- Step 3: He turns right again from the West direction (which means he faces North) and walks 5 km North.
- Step 4: He turns left from the North direction (which means he faces West) and walks 5 km West.
Calculating Net Displacement
To find the shortest distance, we calculate the net displacement in the North-South and East-West directions. Let's consider South and West as negative directions and North and East as positive directions for simplicity, or track them separately.
North-South Displacement:
- Movement South: $5$ km
- Movement North: $5$ km
- Net North-South Displacement = $5 \text{ km (South)} - 5 \text{ km (North)} = 0 \text{ km}$.
This means Rakesh ends up at the same North-South level as his home.
East-West Displacement:
- Movement West: $2$ km
- Movement West: $5$ km
- Net East-West Displacement = $2 \text{ km (West)} + 5 \text{ km (West)} = 7 \text{ km (West)}$.
So, Rakesh's final position is $7$ km directly West of his home.
Determining the Shortest Distance
The shortest distance between two points is a straight line. Since Rakesh's net North-South displacement is $0$ km, his final position is horizontally aligned with his home. The entire displacement is along the East-West axis.
Therefore, the shortest distance between Rakesh's final position and his home is the net East-West displacement.
Shortest Distance = Net East-West Displacement = $7$ km.
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 ?
