Airfoil Lift and Circulation

In aerodynamic wing theory, the lift force generated by air flowing past an aircraft wing profile is fundamentally linked to fluid circulation. According to the foundational Kutta-Joukowski Theorem in aerospace engineering, the aerodynamic lift force vector per unit wing span ($\vec{F}_L$) is directly proportional to the cross product of the free-stream airflow velocity vector ($\vec{v}$) and the bound circulation vector ($\vec{\Gamma}$):

where $\rho$ is the ambient air density.

Consider an aircraft flying through cruising altitude where the air density is $\rho = 1.2\text{ kg/m}^3$. The relative wind velocity flowing horizontally past the wing chord is $\vec{v} = \langle 150, 0, 0 \rangle\text{ m/s}$, and the bound circulation around the airfoil profile is oriented along the spanwise direction as $\vec{\Gamma} = \langle 0, 4, 0 \rangle\text{ m}^2/\text{s}$.

1. Compute the cross product $\vec{v} \times \vec{\Gamma}$.
2. Determine the resulting lift force vector per unit span ($\vec{F}_L$) in Newtons per meter ($\text{N/m}$).
3. State the spatial direction in which the lift force acts.