The couple starts their journey from a common point, which we can assume as the origin of our coordinate system for clarity.
- Husband drives 3 km South from their home, which we denote as moving 3 units down the y-axis from the origin. His position becomes \((0, -3)\).
- Then, he turns 2 km to his right, which is towards the West, moving 2 units on the negative x-axis from his current position. Thus, his final position is \((-2, -3)\).
Now, look at the wife's journey:
- Wife moves 2 km towards the East from their home, which is 2 units on the positive x-axis. Therefore, her position becomes \((2, 0)\).
To find the distance and direction of the wife's office relative to the husband's office:
- The relative position can be found by subtracting the husband's final coordinates from the wife's coordinates. This gives: \[ \text{Relative position} = (2 - (-2), 0 - (-3)) = (4, 3) \]
- The distance is calculated using the distance formula: \[ \text{Distance} = \sqrt{(4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \text{ km} \]
- The direction can be inferred as North-East, considering the relative positive x-axis direction (East) and positive y-axis direction (North).
Thus, the wife's office is located 5 km to the North-east of the husband's office.
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?
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.
