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.