Sharad starts from Point Y and drives 9 km towards the South. He then takes a left turn, drives 19 km, turns right and drives 23 km. He then takes a right turn and drives 8 km. He takes a right turn and drives 17 km. He then turns left, drives 11 km, turns left and drives 6 km to stop at Point Z. How far (shortest distance) and towards which direction should he drive in order to reach Point Y again? (All turns are 90 degrees turns only unless specified)
21 km towards the north
Solving Directional Movement: Point Y to Point Z Journey
This problem involves tracking a series of movements in different directions to find the final displacement. We need to determine the shortest distance and direction from the final point (Point Z) back to the starting point (Point Y).
Tracking Sharad's Movement Details
Let's track Sharad's position step-by-step. We can use a coordinate system where Point Y is the origin (0, 0). We'll consider North as the positive y-axis, South as the negative y-axis, East as the positive x-axis, and West as the negative x-axis.
| Step | Direction of Movement | Distance (km) | Change in X | Change in Y | Position (X, Y) | Current Facing Direction |
| Start | - | - | 0 | 0 | (0, 0) (Point Y) | - |
| 1 | South | 9 | 0 | -9 | (0, -9) | South |
| 2 | East (Left Turn) | 19 | +19 | 0 | (19, -9) | East |
| 3 | South (Right Turn) | 23 | 0 | -23 | (19, -32) | South |
| 4 | West (Right Turn) | 8 | -8 | 0 | (11, -32) | West |
| 5 | North (Right Turn) | 17 | 0 | +17 | (11, -15) | North |
| 6 | West (Left Turn) | 11 | -11 | 0 | (0, -15) | West |
| 7 | South (Left Turn) | 6 | 0 | -6 | (0, -21) (Point Z) | South |
Calculating Net Displacement to Return to Point Y
The final position is Point Z at coordinates (0, -21). The starting position was Point Y at (0, 0).
To return from Point Z to Point Y, we need to calculate the required change in coordinates:
- East-West Displacement (\(\Delta x\)): Required change in x = x-coordinate of Y - x-coordinate of Z = \(0 - 0 = 0\) km.
- North-South Displacement (\(\Delta y\)): Required change in y = y-coordinate of Y - y-coordinate of Z = \(0 - (-21) = +21\) km.
A displacement of \(+21\) km in the y-direction means moving 21 km towards the North.
Final Distance and Direction Calculation
The journey back from Point Z to Point Y requires a net movement of 0 km horizontally (East-West) and 21 km vertically (North).
The shortest distance is calculated using the Pythagorean theorem, based on the net displacements:
Shortest Distance \(= \sqrt{(\Delta x)^2 + (\Delta y)^2}\)
Shortest Distance \(= \sqrt{(0 \text{ km})^2 + (21 \text{ km})^2}\)
Shortest Distance \(= \sqrt{0 + 441 \text{ km}^2}\)
Shortest Distance \(= \sqrt{441 \text{ km}^2}\)
Shortest Distance \(= 21 \text{ km}\)
The direction associated with the net displacement of +21 km in the y-axis is North.
Therefore, to reach Point Y again from Point Z, Sharad should drive 21 km towards the North.
Similar Questions
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Important Questions from Direction and Distance
What is the direction of U with respect to P?
What is the shortest distance between R and T?
If a person walks 3m towards the North from point V, takes a right turn and walks for another 15m, which of the following points would he reach?
In which direction is Amit facing at point F?
What is the distance between the starting point and the end point?
