Consider an airplane in motion in $\mathbb{R}^3$ with velocity $v(t)$ in $\mathbb{R}^3$ relative to surrounding steady air and $\|v(t)\| = v_0 > 0 $ for $t$ in $[0,T]$. Assume that a wind blows with a fixed velocity $w_0 = \|w_0\|$. Assume also that the standard inner product between $w_0$ and $(1,-1,0)$ is negative. Let $x: [0,T] \to \{(x,y,z)\mid x-y=1\}$ be a trajectory of the airplane. Note that $dx/dt$ is not $v(t)$ in general because $x$ represents the actual position of the airplane on the plane with the wind present and $v(t)$ is the velocity of the airplane without wind.
a) What are the conditions on $w_0$ and $v_0$ for a trajectory $x$ to be possibly closed, i.e, $x(0) = x(T)$?
b) What is the maximum area enclosed by a closed trajectory of thie airplane for $[0,T]$?
I tried to solve this long long area maximizing problem, but I couldn't solve it. Please try solve this. It's too difficult to me.