Mapping Vinay's Journey to Find the Return Path

This problem involves tracking a person's movement across different directions and calculating the shortest distance and direction to return to the starting point. We can solve this by visualizing the path or using a coordinate system.

Understanding Vinay's Movements

Let's break down Vinay's journey step by step:

• Start at Point A. We can consider Point A as the origin (0, 0) on a coordinate plane, where North is the positive y-axis and East is the positive x-axis.
• Drives 3 km North. Vinay moves from (0, 0) to (0, 3).
• Takes a right turn and drives 2 km. A right turn from North is East. So, he moves 2 km East. His new position is (0 + 2, 3) = (2, 3).
• Turns left and drives 3 km. A left turn from East is North. He moves 3 km North. His new position is (2, 3 + 3) = (2, 6).
• Takes a left turn and drives 6 km. A left turn from North is West. He moves 6 km West. His new position is (2 - 6, 6) = (-4, 6).
• Takes a final left turn, drives 6 km. A left turn from West is South. He moves 6 km South. His final position is (-4, 6 - 6) = (-4, 0).
• Stops at Point P. So, Point P is located at coordinates (-4, 0).

Calculating the Distance and Direction from P to A

Now, we need to find the shortest distance and direction from Vinay's final location (Point P) back to his starting point (Point A).

• Point P coordinates: (-4, 0)
• Point A coordinates: (0, 0)

To get from Point P (-4, 0) to Point A (0, 0), Vinay needs to change his position:

• Change in the x-coordinate: $0 - (-4) = +4$ km. A positive change in the x-coordinate means moving towards the East.
• Change in the y-coordinate: $0 - 0 = 0$ km. There is no change in the North-South direction.

The net displacement required is 4 km in the East direction.

Determining the Shortest Distance

The shortest distance between two points in a coordinate plane is a straight line. We can use the distance formula, which is derived from the Pythagorean theorem:

Distance = $\sqrt{(\Delta x)^2 + (\Delta y)^2}$

Substituting the changes we found:

Distance = $\sqrt{(4)^2 + (0)^2}$

Distance = $\sqrt{16 + 0}$

Distance = $\sqrt{16}$

Distance = $4$ km

Final Answer

Vinay needs to travel 4 km towards the East to reach Point A again.