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Question

A tourist drives $20 \text{ km}$ towards east, turns right and drives $6 \text{ km}$, then drives $6 \text{ km}$ towards west. He then turns to his left and drives $4 \text{ km}$ and finally turns right and drives $14 \text{ km}$. Where is he from his starting point?

The correct answer is

$10 \text{ km}$ towards south

Directional Movement Calculation

This problem requires calculating the final displacement of a tourist from their starting point after a series of directional movements.

Tracking Tourist Movement Steps

We can track the tourist's position step-by-step using a coordinate system. Assume the starting point is the origin $(0, 0)$, East is the positive x-axis, and North is the positive y-axis.

  • Initial Move: The tourist drives $20 \text{ km}$ East. The position becomes $(20, 0)$. The direction faced is East.
  • First Turn: The tourist turns right from East, now facing South. They drive $6 \text{ km}$ South. The position becomes $(20, -6)$. The direction faced is South.
  • Westward Travel: The tourist drives $6 \text{ km}$ West. The position changes by $(-6, 0)$. The new position is $(20 - 6, -6) = (14, -6)$. The direction faced is West.
  • Second Turn: The tourist turns left from West, now facing South. They drive $4 \text{ km}$ South. The position changes by $(0, -4)$. The new position is $(14, -6 - 4) = (14, -10)$. The direction faced is South.
  • Final Turn: The tourist turns right from South, now facing West. They drive $14 \text{ km}$ West. The position changes by $(-14, 0)$. The new position is $(14 - 14, -10) = (0, -10)$.

Final Position Analysis

The final coordinates are $(0, -10)$. This indicates a net movement of 0 km in the East-West direction and 10 km in the South direction relative to the starting point $(0, 0)$.

Therefore, the tourist is located $10 \text{ km}$ towards South from their starting point.

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Important Questions from Direction and Distance (Notes)

  1. A woman leaves her home. She walks 40 m in North-West direction and then 90 m in South-East direction. Then, she moves 30 m in North direction. How far is she now from her initial position ?
  2. N sits to the North of L and West of M. L sits to the East of O. If everyone is facing North, which direction of O, is N?
  3. I starts from Point A and drives 5 km towards the north. He then takes a right turn, drives 2 km, turns right and drives 9 km. He then takes a right turn and drives 7 km. He takes a final right turn, drives 4 km and stops at Point P. How far (shortest distance) and towards which direction should he drive in order to reach Point A again? (All turns are $90^\circ$ turns only unless specified.)

  4. A Couple started to their offices in the morning from their home. While husband drives 3 km South and then 2 km to his right to reach his office, wife goes 2 km towards East for her office. With respect to husband's office, how far and in which direction is the wife's office ?
  5. Point A is $30$ m to the North of point B. Point A is $10$ m to the west of point C. Point C is $20$m to the North of point D. Point E is $20$ m to the East of point D. Point F is $20$ m to the South of point E. What is the shortest distance from the point B to point E.

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