Understanding Sanjeev's Journey
This problem involves tracking the movement of Sanjeev in different directions and calculating his final position relative to his starting point. We need to find both the net distance and the direction.
Step-by-Step Movement Analysis
Let's break down Sanjeev's journey step by step:
- Initial Move: Sanjeev travels 3 km towards the East.
- Second Move: He turns South and travels 3 km.
- Third Move: He turns East again and travels 6 km.
- Final Move: He turns North and travels 15 km.
Calculating Net East-West Displacement
We first sum up all the movements along the East-West axis. Movement towards the East is considered positive, and West is negative.
- Movement East: 3 km + 6 km = 9 km
- Movement West: 0 km
- Net Horizontal Displacement: 9 km towards the East.
Calculating Net North-South Displacement
Next, we sum up all the movements along the North-South axis. Movement towards the North is considered positive, and South is negative.
- Movement North: 15 km
- Movement South: 3 km
- Net Vertical Displacement: 15 km (North) - 3 km (South) = 12 km towards the North.
Determining Final Distance from Starting Point
Sanjeev's final position is 9 km East and 12 km North of his starting point. These two displacements form the legs of a right-angled triangle. The distance from the starting point is the hypotenuse of this triangle. We can use the Pythagorean theorem to calculate this distance:
Let $d$ be the net distance from the starting point.
According to the Pythagorean theorem: $d^2 = (\text{Net Horizontal Displacement})^2 + (\text{Net Vertical Displacement})^2$
Substituting the values:
$$ d^2 = (9 \text{ km})^2 + (12 \text{ km})^2 $$ $$ d^2 = 81 \text{ km}^2 + 144 \text{ km}^2 $$ $$ d^2 = 225 \text{ km}^2 $$ $$ d = \sqrt{225 \text{ km}^2} $$ $$ d = 15 \text{ km} $$Determining Final Direction from Starting Point
Since the net horizontal displacement is towards the East (9 km) and the net vertical displacement is towards the North (12 km), Sanjeev's final position relative to his starting point is in the North-East direction.
Conclusion
Combining the distance and direction, Sanjeev is 15 km away from his starting point in the North-East direction.
Important Questions from Direction and Distance
- Vijendra walks a certain distance, say X metres, from his home towards the west. Then he turns left and walks 23 metres. After that, he turns left and walks 36 metres. Then he turns left again to walk 23 metres. He finally turns left and walks 18 metres to reach his home. Find the value of X.
Gaurav exits from the backdoor of his north-facing house and walks 25 m straight, then he takes a left turn and walks 36 m, then he turns left and walks 47 m. He turns left again and walks 36 m. How far and in which direction is he from his house now?
Reeta is standing facing south-east. First, she turns 135° clockwise. After that, she turns 90 ° anticlockwise. Then she turns 45 ° clockwise, followed by a 135 ° anticlockwise turn. In which direction is she facing now?
Sahasra started running from her house towards the north. After 40 meters she turned left and ran for 75 meters. She then turned right and ran for 30 meters, and again turned right to run 75 meters. In which direction was she running finally?
At the time of sunset, Lopa and Kritika are sitting facing each other. If the shadow of Lopa falls to the right of Kritika, in which direction is Kritika facing?
