$P$, $Q$ and $R$ live in the same city. $R$ 's home is $21\text{ km}$ East of $Q$ 's home and $Q$ 's home is $20\text{ km}$ South of $P$ 's home. In which direction is $P$ 's home located with respect to $R$ 's home?

Let's map the locations on a standard coordinate system with $Q$ 's home at the reference origin $(0,0)$:
1. "$R$ 's home is $21\text{ km}$ East of $Q$ 's home":
Coordinates of $R=(21,0)$
2. "$Q$ 's home is $20\text{ km}$ South of $P$ 's home" means $P$ 's home is $20\text{ km}$ North of $Q$ 's home:
Coordinates of $P=(0,20)$

Now, find the relative direction of $P(0,20)$ from the perspective of $R(21,0)$:

* Moving from $R$ to $P$ requires moving leftwards (West) from $x=21$ to $x=0$.
* It also requires moving upwards (North) from $y=0$ to $y=20$.

Combining these components, $P$ 's home is located in the North-West direction with respect to $R$ 's home.