Vasudha started jogging from a point near her house. She first jogs $3\text{km}$ towards West, then she turned towards North and moved $4\text{km}$ in that direction. What is the shortest distance and in which direction is Vasudha from her house?

Let Vasudha's house be at the origin $(0,0)$.
1. She jogs $3\text{km}$ West: Her position becomes $(-3,0)$.
2. She turns North and moves $4\text{km}$: Her final position becomes $(-3,4)$.

To find the shortest distance from the house $(0,0)$ to $(-3,4)$, we apply the Pythagorean theorem:

$$\text{Shortest Distance}=\sqrt{(-3{)}^2+4^2}=\sqrt{9+16}=\sqrt{25}=5\text{km}$$

Relative to the starting point $(0,0)$, the point $(-3,4)$ lies in the North-West quadrant.

Therefore, she is $5\text{km}$, North-West from her house.