A man is facing the west direction and from there he walks $10\text{km}$ in the right and then turns left and walks another $6\text{km}$. Then, he turns $90^{\circ }$ clockwise and walks for another $6\text{km}$ and finally walks $6\text{km}$ in the east. Find the distance between his initial and final position.

Let's plot the movements step-by-step starting from an initial origin point $O(0,0)$:The man is facing West and walks $10\text{km}$ to his right. Turning right while facing West means he heads North.Position after step 1: $10\text{km}$ North from origin. $(0,10)$ Then he turns left and walks $6\text{km}$. Turning left from North means he heads West.Position after step 2: $6\text{km}$ West and $10\text{km}$ North. $(-6,10)$ Then he turns $90^{\circ }$ clockwise and walks $6\text{km}$. Turning clockwise from West means he heads North.Position after step 3: $(-6,10+6)=(-6,16)$ Finally, he walks $6\text{km}$ towards the East.Position after step 4: $(-6+6,16)=(0,16)$ Now let's calculate the shortest straight-line distance between the initial position $(0,0)$ and the final position $(0,16)$:
$$\text{Distance}=16-0=16\text{km}$$
The final position lies exactly $16\text{km}$ North of the starting point.