Mapping Vinay's Journey to Find the Return Path
This problem involves tracking a person's movement across different directions and calculating the shortest distance and direction to return to the starting point. We can solve this by visualizing the path or using a coordinate system.
Understanding Vinay's Movements
Let's break down Vinay's journey step by step:
- Start at Point A. We can consider Point A as the origin (0, 0) on a coordinate plane, where North is the positive y-axis and East is the positive x-axis.
- Drives 3 km North. Vinay moves from (0, 0) to (0, 3).
- Takes a right turn and drives 2 km. A right turn from North is East. So, he moves 2 km East. His new position is (0 + 2, 3) = (2, 3).
- Turns left and drives 3 km. A left turn from East is North. He moves 3 km North. His new position is (2, 3 + 3) = (2, 6).
- Takes a left turn and drives 6 km. A left turn from North is West. He moves 6 km West. His new position is (2 - 6, 6) = (-4, 6).
- Takes a final left turn, drives 6 km. A left turn from West is South. He moves 6 km South. His final position is (-4, 6 - 6) = (-4, 0).
- Stops at Point P. So, Point P is located at coordinates (-4, 0).
Calculating the Distance and Direction from P to A
Now, we need to find the shortest distance and direction from Vinay's final location (Point P) back to his starting point (Point A).
- Point P coordinates: (-4, 0)
- Point A coordinates: (0, 0)
To get from Point P (-4, 0) to Point A (0, 0), Vinay needs to change his position:
- Change in the x-coordinate: $0 - (-4) = +4$ km. A positive change in the x-coordinate means moving towards the East.
- Change in the y-coordinate: $0 - 0 = 0$ km. There is no change in the North-South direction.
The net displacement required is 4 km in the East direction.
Determining the Shortest Distance
The shortest distance between two points in a coordinate plane is a straight line. We can use the distance formula, which is derived from the Pythagorean theorem:
Distance = $\sqrt{(\Delta x)^2 + (\Delta y)^2}$
Substituting the changes we found:
Distance = $\sqrt{(4)^2 + (0)^2}$
Distance = $\sqrt{16 + 0}$
Distance = $\sqrt{16}$
Distance = $4$ km
Final Answer
Vinay needs to travel 4 km towards the East to reach Point A again.
Important Questions from Direction and Distance
Sita took an auto from her home in Andheri (A) to go to her college in Fatehpuri (F). Rather than continuing straight on the direct road to the college that had no turns, the auto driver took a diversion after 10 km and tumed right at Bandra (B) crossing, then at Colaba T- point (C) after 8 km turned left, again after covering 12 km turned left at Dalhousi Building (D) and soon after 8 km turned right at Elphinston point (E) and after covering 4 km reached Fatehpuri (F). Had the auto driver taken the direct route how much less distance would Sita have actually travelled between the starting point and the destination?
Radha walks a distance of 9 m towards the South-East. Then she walks 15 m towards the West. From here, she walks 9 m towards the North-West. Finally she walks 6 m towards the East and stands at the point. How far is she standing from the starting point?
According to the time by Nihaal's watch its half past one and the hour hand is pointing towards the north-east. Assuming that there is no change in Nihaal's position, in which direction would the minute hand point after 30 minutes?
According to the time by Rohan's watch, it is half past 6 and the watch's hands are pointing to the south. In which of the given directions will the minute-hand point AFTER exactly 24 hours?
According to the time by Rick's watch its quarter past eight. If the minute hand points towards the east, towards which direction would the hour hand point?
