I.10.27 Flux of the vector field

Condition: Find the flow of the vector field \( \vec{a}=(x-2 z) \vec{\imath}+(x+3 y+z) \vec{\jmath}+(5 x+y) \vec{k} \) through the upper side of the triangle \( A B C \) cut off by the coordinate planes from the plane \( \alpha x+\beta y+\gamma z=1 \). Solve the problem in two ways: conventional and through projections. \[ \alpha=-3, \quad \beta=2, \quad \gamma=4 \]