In the deduction of boundary conditions of electric field we take a surface and do calculations mathematically.enter image description here

Why do we take the direction of E perpendicular and parallel in same direction across two surface? How can the deduction based on this assertion be used as a general one. ? What will happen if the direction are arbitrary.? Deriving the Electrostatic boundary conditions This link question's last part is related to mine, i just want to know why this specific orientation is taken as it won't be a general one?