Detailed Solution for Hari's Driving Distance and Direction

This problem involves tracking a person's movement through a series of directions and distances to determine the final displacement needed to return to the starting point. We can solve this by mapping Hari's path using a coordinate system.

Mapping Hari's Path Step-by-Step

Let's assume Hari starts at Point A, which we can represent as the origin (0, 0) on a coordinate plane. We'll track his position after each leg of the journey.

• Start at Point A: (0, 0)
• 1. Drives 10 km West: Moving west decreases the x-coordinate. New position: $(-10, 0)$.
• 2. Takes a right turn (North) and drives 8 km: Moving north increases the y-coordinate. New position: $(-10, 0 + 8) = (-10, 8)$.
• 3. Turns right (East) and drives 17 km: Moving east increases the x-coordinate. New position: $(-10 + 17, 8) = (7, 8)$.
• 4. Turns right (South) and drives 12 km: Moving south decreases the y-coordinate. New position: $(7, 8 - 12) = (7, -4)$.
• 5. Takes a final right turn (West) and drives 7 km: Moving west decreases the x-coordinate. New position: $(7 - 7, -4) = (0, -4)$. This is the final location, Point P.

Calculating the Return Trip to Point A

Hari stops at Point P, which is located at coordinates (0, -4). His starting point, A, is at (0, 0).

To find the shortest distance and direction from Point P to Point A, we compare their coordinates:

• Current position (P): $(0, -4)$
• Starting position (A): $(0, 0)$

We calculate the difference in the coordinates:

• Difference in East-West (x-axis): $0 - 0 = 0$ km. This means Hari is directly North or South of his starting point A.
• Difference in North-South (y-axis): $0 - (-4) = +4$ km. The positive value indicates that Point A is 4 km North of Point P's y-coordinate.

The displacement vector from P to A is $(0, 4)$.

The shortest distance is the magnitude of this displacement vector:

Distance $= \sqrt{(\Delta x)^2 + (\Delta y)^2} = \sqrt{(0)^2 + (4)^2} = \sqrt{0 + 16} = \sqrt{16} = 4$ km.

Since the change in the y-coordinate ($\Delta y$) is positive ($+4$), the direction from Point P to Point A is North.

Final Answer Explanation

Therefore, Hari needs to drive 4 km to the north from Point P to reach Point A again.