Why is the perpendicular part a scalar in the electrodynamic boundary conditions at surfaces?

Question

The general boundary conditions of electrodynamics is given as follows: $$D_{1}^{\perp}-D_{2}^{\perp} = \sigma_{f},~~ B_{1}^{\perp} - B_{2}^{\perp} = 0, ~\vec{E}_{1}^{\parallel} - \vec{E}_{2}^{\parallel} = 0, \vec{H}_{1}^{\parallel}- \vec{H}_{2}^{\parallel} = \vec{K}_{f} \times \hat{n}$$

Question: Is there a reason why whenever boundary conditions are given, the amount of discontinuity of the parallel components of the fields is given as a scalar quantity whereas the parallel components are given as vectors? Is this because there are a number of ways for a vector field to be parallel to a surface (which is a assumed flat) but only a unique vector field describes a perpendicular field to a surface (other than the sign)?

Answer 1

A surface in 3D has a two-dimensional tangent plane but only a one-dimensional normal line at each point. So the part of the field normal to the surface can be specified by just a single number, essentially the coefficient in front of the unique normal unit vector, while the part parallel is still a two-dimensional vector.