Understanding the Direction and Distance Problem

This question asks us to find the straight-line distance between a starting point 'P' and an ending point 'Q' after a series of movements in different directions. We need to trace the path taken by Rohit step-by-step to determine the final position relative to the starting point.

Step-by-Step Solution for Distance Calculation

Let's break down Rohit's walk into individual segments and track his position relative to the starting point 'P'. We can consider East as positive in the horizontal direction and North as positive in the vertical direction.

Analyzing Rohit's Movements from Point P

• Movement 1: From point P, Rohit walks 5 km towards the east.
• Movement 2: He then turns right. When facing East, a right turn means turning towards South. He walks for 2 km in the south direction.
• Movement 3: Again he turns right. When facing South, a right turn means turning towards West. He walks for 3 km in the west direction.
• Movement 4: He then again turns right. When facing West, a right turn means turning towards North. He walks for 2 km in the north direction to reach point 'Q'.

Calculating the Net Displacement

We can sum up the movements in each direction to find the overall change in position from P to Q.

• East-West Movement: Rohit moved 5 km East and then 3 km West. The net movement in the East-West direction is \(5 \text{ km (East)} - 3 \text{ km (West)} = 2 \text{ km East}\).
• North-South Movement: Rohit moved 2 km South and then 2 km North. The net movement in the North-South direction is \(2 \text{ km (North)} - 2 \text{ km (South)} = 0 \text{ km}\).

So, point Q is located 2 km to the East and 0 km to the North or South of point P.

Determining the Distance Between P and Q

Since point Q is exactly 2 km East and 0 km North/South from point P, the straight-line distance between P and Q is simply the net displacement in the East-West direction.

Distance PQ = Net Eastward displacement

Distance PQ = \(2 \text{ km}\)

We can visualize this path. Starting at P, he moves 5 East. Then 2 South. Then 3 West (which brings him back 3 units towards the starting East-West line). Then 2 North (which brings him back up to the starting North-South line). His final position is 2 km East of the starting point P, and at the same North-South level as P.

Conclusion

Based on the analysis of Rohit's movements, the distance between the starting point 'P' and the ending point 'Q' is 2 km.

Net displacement East = \(5 \text{ km} - 3 \text{ km} = 2 \text{ km East}\)

Net displacement North = \(2 \text{ km} - 2 \text{ km} = 0 \text{ km}\)

The final position Q is 2 km East and 0 km North/South from P. The distance PQ is the straight-line distance, which is 2 km.

Revision Table: Key Concepts

Additional Information: Directional Problems

Direction and distance problems are common in reasoning tests. They typically involve tracing a path based on directions (North, South, East, West) and turns (left, right).

• Tracing the Path: It's often helpful to draw a simple diagram to visualize the movements.
• Net Displacement: Instead of just total distance walked, these problems usually ask for the straight-line distance from the start or the final direction from the start. This requires calculating the net displacement in the horizontal (East-West) and vertical (North-South) directions.
• Pythagorean Theorem: If there is a net displacement in both the horizontal and vertical directions, the straight-line distance can be found using the Pythagorean theorem \(d = \sqrt{(\Delta x)^2 + (\Delta y)^2}\), where \(\Delta x\) is the net horizontal displacement and \(\Delta y\) is the net vertical displacement. In this specific problem, the net vertical displacement is 0, simplifying the calculation.
• Turns: Remember that 'right' and 'left' turns are relative to the current direction of movement.

Solving these problems systematically by breaking down movements into cardinal directions makes them easier to manage.