Mr. Yanki's Driving Problem: Finding the Return Path
This problem involves tracking movement based on directions and distances to find the shortest path back to the starting point. We can solve this by treating the movements as vectors or by using a coordinate system.
Understanding Mr. Yanki's Movements
Let's break down each step of Mr. Yanki's journey. We'll use a coordinate system where Point A is the origin (0, 0). We'll assume North corresponds to the positive y-axis and East corresponds to the positive x-axis. Therefore, South is the negative y-axis and West is the negative x-axis.
Tracking Position with Coordinates
We can track Mr. Yanki's position after each segment of his journey. A table helps visualize these changes.
| Movement Segment | Direction Driven | Distance (km) | Turn Made | Facing Direction | Coordinate Change (Δx, Δy) | Current Position (x, y) |
|---|---|---|---|---|---|---|
| Start (Point A) | - | - | - | - | (0, 0) | (0, 0) |
| 1 | South | 17 | - | South | (0, -17) | (0, -17) |
| 2 | West | 19 | Right | West | (-19, 0) | (-19, -17) |
| 3 | North | 29 | Right | North | (0, 29) | (-19, 12) |
| 4 | East | 15 | Right | East | (15, 0) | (-4, 12) |
| 5 | South | 1 | Right | South | (0, -1) | (-4, 11) |
| 6 | East | 6 | Left | East | (6, 0) | (2, 11) |
| 7 (Stop at P) | South | 11 | Right | South | (0, -11) | (2, 0) |
Calculating Final Position
After all the movements, Mr. Yanki stops at Point P. Let's calculate the net displacement in the x (East-West) and y (North-South) directions:
- Net East-West Displacement: Sum of x-changes = $0 + (-19) + 0 + 15 + 0 + 6 + 0 = 2$ km. This means Point P is 2 km East of Point A.
- Net North-South Displacement: Sum of y-changes = $(-17) + 0 + 29 + 0 + (-1) + 0 + (-11) = 17 - 17 = 0$ km. This means Point P is on the same East-West line as Point A.
So, the final position of Point P relative to Point A is (2, 0).
Determining Return Path to Point A
Mr. Yanki is currently at Point P (2, 0) and needs to return to Point A (0, 0).
The required displacement vector from P to A is:
Displacement = (Target x - Current x, Target y - Current y)
Displacement = ($0 - 2$, $0 - 0$) = (-2, 0)
A coordinate change of (-2, 0) means moving 2 units in the negative x-direction.
- The negative x-direction corresponds to the West direction.
- The magnitude of the movement is $ \sqrt{(-2)^2 + (0)^2} = \sqrt{4} = 2 $ km.
Therefore, Mr. Yanki needs to drive 2 km towards the West to reach Point A again.
Important Questions from Direction and Distance
- Vijendra walks a certain distance, say X metres, from his home towards the west. Then he turns left and walks 23 metres. After that, he turns left and walks 36 metres. Then he turns left again to walk 23 metres. He finally turns left and walks 18 metres to reach his home. Find the value of X.
Gaurav exits from the backdoor of his north-facing house and walks 25 m straight, then he takes a left turn and walks 36 m, then he turns left and walks 47 m. He turns left again and walks 36 m. How far and in which direction is he from his house now?
Reeta is standing facing south-east. First, she turns 135° clockwise. After that, she turns 90 ° anticlockwise. Then she turns 45 ° clockwise, followed by a 135 ° anticlockwise turn. In which direction is she facing now?
Sahasra started running from her house towards the north. After 40 meters she turned left and ran for 75 meters. She then turned right and ran for 30 meters, and again turned right to run 75 meters. In which direction was she running finally?
At the time of sunset, Lopa and Kritika are sitting facing each other. If the shadow of Lopa falls to the right of Kritika, in which direction is Kritika facing?
