Answer
The direction of the wind is $~~22.2^{\circ}~~$ south of west.
Work Step by Step
Let $v_{pg}$ be the velocity of the plane relative to the ground.
Let $v_{pa}$ be the velocity of the plane relative to the air.
Let $v_{ag}$ be the velocity of the air relative to the ground.
We can find the east-west component of the velocity of the air relative to the ground:
$v_{pg,x} = v_{pa,x}+v_{ag,x}$
$v_{ag,x} = v_{pg,x} - v_{pa,x}$
$v_{ag,x} = 0 - (500~km/h)~sin~20.0^{\circ}~(east)$
$v_{ag,x} = -171~km/h~(east)$
$v_{ag,x} = 171~km/h~(west)$
We can find the north-south component of the velocity of the air relative to the ground:
$v_{pg,y} = v_{pa,y}+v_{ag,y}$
$v_{ag,y} = v_{pg,y} - v_{pa,y}$
$v_{ag,y} = (400~km/h) - (500~km/h)~cos~20^{\circ}$
$v_{ag,y} = -69.8~km/h ~(north)$
$v_{ag,y} = 69.8~km/h ~(south)$
We can find the direction of the wind as an angle $\theta$ that is south of west:
$tan~\theta = \frac{69.8~km/h}{171~km/h}$
$\theta = tan^{-1}~(\frac{69.8~km/h}{171~km/h})$
$\theta = 22.2^{\circ}$
The direction of the wind is $~~22.2^{\circ}~~$ south of west.
