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.)
This problem involves calculating the shortest distance and direction required to return to the starting point after a series of movements. We can solve this by tracking the position using a coordinate system.
Movement Tracking
Let's assume the starting Point A is at the origin (0, 0) of a coordinate plane, where North corresponds to the positive y-axis and East corresponds to the positive x-axis.
The movements are as follows:
- Starts at A (0, 0).
- Drives 5 km towards the north. New position: (0, 5).
- Takes a right turn (now facing East) and drives 2 km. New position: (0+2, 5) = (2, 5).
- Turns right (now facing South) and drives 9 km. New position: (2, 5-9) = (2, -4).
- Turns right (now facing West) and drives 7 km. New position: (2-7, -4) = (-5, -4).
- Takes a final right turn (now facing North) and drives 4 km. New position: (-5, -4+4) = (-5, 0).
- Stops at Point P (-5, 0).
| Step | Direction | Distance (km) | Coordinate Change | Position |
|---|---|---|---|---|
| 1 | North | 5 | (0, +5) | (0, 5) |
| 2 | East | 2 | (+2, 0) | (2, 5) |
| 3 | South | 9 | (0, -9) | (2, -4) |
| 4 | West | 7 | (-7, 0) | (-5, -4) |
| 5 | North | 4 | (0, +4) | (-5, 0) |
Final Position Calculation
The final position, Point P, is located at coordinates (-5, 0).
Distance and Direction Calculation
We need to find the shortest distance and direction from Point P (-5, 0) back to Point A (0, 0).
- Horizontal Displacement: The change in the x-coordinate is $0 - (-5) = +5$ km. A positive change in x means movement towards the East.
- Vertical Displacement: The change in the y-coordinate is $0 - 0 = 0$ km. There is no net change in the North-South direction.
The shortest distance can be calculated using the distance formula, which is equivalent to finding the magnitude of the displacement vector:
Distance $= \sqrt{(\Delta x)^2 + (\Delta y)^2} = \sqrt{(5)^2 + (0)^2} = \sqrt{25} = 5$ km.
Since the displacement is entirely along the positive x-axis (+5 km), the direction needed to travel from Point P to Point A is East.
Therefore, one needs to drive 5 km to the east to reach Point A again from Point P.
Important Questions from Direction and Distance (Notes)
- If a map is placed in such a manner that east becomes southeast, then what will northeast become?
Point A is $30$ m to the North of point B. Point A is $10$ m to the west of point C. Point C is $20$m to the North of point D. Point E is $20$ m to the East of point D. Point F is $20$ m to the South of point E. What is the shortest distance from the point B to point E.
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.
- The door of Aman's house faces east. From the back side of his house, Aman walks straight 30m. Again he walks 20m after turning to his left, then he turns to his right and walks 30m. Finally he turn towards north and walks 20m and stops. So, what is the distance between the starting and the end point?
- Smitha runs 10 km south from her flat, turns left and walks 23 km again turns left and walks 40 km then turns right and walks 5 km to reach her office. In which direction is the office from her house?
