Understanding Kamal's Driving Path

Kamal starts from his house and follows a specific sequence of movements. We need to determine the direction he is driving when he turns back towards his house after the last leg of his journey.

Step-by-Step Direction Analysis

1. Initial Movement: Kamal drives 10 km in the North-west direction. This moves him North and West from his house.
2. Second Movement: He then drives 10 km in the South-west direction. This movement adds further Westward displacement and cancels out the Northward displacement from the first step, placing him directly West of his house.
3. Third Movement: Next, he drives 10 km in the South-east direction. This movement cancels out some of the Westward displacement and adds Southward displacement.

Determining the Final Direction Home

Let's analyze the position using coordinates, assuming the house is at the origin (0,0).

• Move 1 (North-west): Displacement vector $ \vec{d_1} = 10 (\cos(135^\circ) \hat{i} + \sin(135^\circ) \hat{j}) = 10 (-\frac{\sqrt{2}}{2} \hat{i} + \frac{\sqrt{2}}{2} \hat{j}) $.
• Move 2 (South-west): Displacement vector $ \vec{d_2} = 10 (\cos(225^\circ) \hat{i} + \sin(225^\circ) \hat{j}) = 10 (-\frac{\sqrt{2}}{2} \hat{i} - \frac{\sqrt{2}}{2} \hat{j}) $.
• Move 3 (South-east): Displacement vector $ \vec{d_3} = 10 (\cos(315^\circ) \hat{i} + \sin(315^\circ) \hat{j}) = 10 (\frac{\sqrt{2}}{2} \hat{i} - \frac{\sqrt{2}}{2} \hat{j}) $.

The total displacement from the house is $ \vec{D} = \vec{d_1} + \vec{d_2} + \vec{d_3} $.
$ \vec{D} = 10 \left( (-\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}) \hat{i} + (\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}) \hat{j} \right) $
$ \vec{D} = 10 \left( -\frac{\sqrt{2}}{2} \hat{i} - \frac{\sqrt{2}}{2} \hat{j} \right) $

Kamal's final position is $ (-\frac{10\sqrt{2}}{2}, -\frac{10\sqrt{2}}{2}) $, which is South-west of his house. To return home (0,0), he must travel in the opposite direction of $ \vec{D} $.
The return direction vector is $ -\vec{D} = -10 \left( -\frac{\sqrt{2}}{2} \hat{i} - \frac{\sqrt{2}}{2} \hat{j} \right) = 10 \left( \frac{\sqrt{2}}{2} \hat{i} + \frac{\sqrt{2}}{2} \hat{j} \right) $.

This vector points in the positive x (East) and positive y (North) directions. Therefore, Kamal is most likely driving in the North-east direction towards his house.