Understanding Johny's Driving Journey

This problem requires us to track Johny's movements from a starting point (Point A) through a series of directional changes and distances, ultimately determining the shortest path back to Point A from his final location (Point P).

Tracking Johny's Movement Step-by-Step

We can map Johny's journey using a 2D coordinate system. Let's set the starting Point A as the origin (0, 0). We'll consider North as the positive y-direction and East as the positive x-direction.

• Start at Point A: Johny begins at coordinates (0, 0).
• Move 1: 6 km North.
  • His position changes by +6 in the y-direction.
  • New position: (0, 0 + 6) = (0, 6).
  • Johny is now facing North.
• Move 2: Right turn, 17 km.
  • A right turn from North makes him face East (positive x-direction).
  • His position changes by +17 in the x-direction.
  • New position: (0 + 17, 6) = (17, 6).
  • Johny is now facing East.
• Move 3: Right turn, 13 km.
  • A right turn from East makes him face South (negative y-direction).
  • His position changes by -13 in the y-direction.
  • New position: (17, 6 - 13) = (17, -7).
  • Johny is now facing South.
• Move 4: Right turn, 8 km.
  • A right turn from South makes him face West (negative x-direction).
  • His position changes by -8 in the x-direction.
  • New position: (17 - 8, -7) = (9, -7).
  • Johny is now facing West.
• Move 5: Right turn, 7 km.
  • A right turn from West makes him face North (positive y-direction).
  • His position changes by +7 in the y-direction.
  • Final position (Point P): (9, -7 + 7) = (9, 0).

Calculating the Return Journey to Point A

Johny's final location, Point P, is at coordinates (9, 0). His starting point, A, is at (0, 0).

To find out how far and in which direction he needs to travel to get from Point P back to Point A, we calculate the displacement vector from P to A:

• Change in x-coordinate (\(\Delta x\)): Final x - Initial x = \(0 - 9 = -9\) km.
• Change in y-coordinate (\(\Delta y\)): Final y - Initial y = \(0 - 0 = 0\) km.

The displacement vector is (-9, 0).

Interpreting this displacement:

• A change of -9 in the x-coordinate means moving 9 km in the negative x-direction. Since positive x is East, negative x is **West**.
• A change of 0 in the y-coordinate means no movement North or South is needed.

Final Distance and Direction Calculation

The shortest distance is the magnitude of the displacement vector:

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

Distance = \(\sqrt{(-9)^2 + (0)^2}\)

Distance = \(\sqrt{81 + 0}\)

Distance = \(\sqrt{81}\)

Distance = 9 km.

The direction is determined by the displacement vector (-9, 0), which indicates movement solely towards the West.

Therefore, Johny needs to drive 9 km towards the West to reach Point A again.