Calculating Final Position from Multiple Directions

This problem requires us to track the displacement of the seller from her starting point after a series of movements in different directions. We can solve this by breaking down the movements into North/South and East/West components.

Step-by-Step Movement Analysis

Let's assume the starting position is the origin (0,0). We can track the changes in position after each step:

1. Starts at: (0, 0)
2. Walks 7 km north: The position changes in the North direction. New position relative to start: (0 km East/West, +7 km North).
3. Turns west and walks 6 km: From the current position (+7 km North), she moves 6 km West. Changes in East/West position: -6 km. New position relative to start: (-6 km West, +7 km North).
4. Turns south and walks 2 km: From the current position (-6 km West, +7 km North), she moves 2 km South. Changes in North/South position: -2 km. New position relative to start: (-6 km West, +7 km - 2 km South) = (-6 km West, +5 km North).
5. Turns west and walks 3 km: From the current position (-6 km West, +5 km North), she moves 3 km West. Changes in East/West position: -3 km. New position relative to start: (-6 km West - 3 km West, +5 km North) = (-9 km West, +5 km North).
6. Turns to her left and walks 5 km: The previous movement was West. A left turn from West means turning South. From the current position (-9 km West, +5 km North), she moves 5 km South. Changes in North/South position: -5 km. New position relative to start: (-9 km West, +5 km North - 5 km South) = (-9 km West, 0 km North/South).

Calculating Total Displacement

Alternatively, we can sum the total displacement in the East/West and North/South directions:

• North movements: +7 km
• South movements: +2 km + 5 km = +7 km (Note: the last movement was South, equivalent to a left turn from West)
• East movements: 0 km
• West movements: +6 km + 3 km = +9 km

Now, let's find the net displacement:

• Net North/South displacement: Total North - Total South = \(\text{+7 km} - \text{7 km} = \text{0 km}\). There is no net displacement in the North or South direction relative to the start.
• Net East/West displacement: Total East - Total West = \(\text{0 km} - \text{9 km} = \text{-9 km}\). The negative sign indicates a displacement towards the West. So, there is a net displacement of 9 km West.

Final Position Relative to Starting Position

Combining the net displacements, the final position of the seller is 9 km West and 0 km North/South relative to her starting position.

This means she is exactly 9 km to the west of where she began.

The final position is 9 km West of the starting point.

Revision Table: Direction and Displacement Concepts

Additional Information: Understanding Displacement vs. Distance

It is important to distinguish between distance and displacement.

• Distance: The total length of the path traveled. In this problem, the total distance is \(7 + 6 + 2 + 3 + 5 = 23\) km.
• Displacement: The shortest straight-line distance and direction from the starting point to the ending point. It is a vector quantity. In this problem, the displacement is 9 km West.

Displacement only depends on the initial and final positions, not the path taken. This problem specifically asks for the position "with reference to her starting position", which means we need to find the displacement.