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)
- 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?
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 ?
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.
