I am therefore studying the orientation of the boundaries (for example, Gulliemin and Pollack, page 97, Mukherjee, page 182) and it is stated that, in the first introduction,

$ dim M geq 1 $ is a smooth variety with limit and since $ codim partial M = $ 1 every $ x in partial M $ there are two orthogonal vectors in the tangent space $ T_x (M) $ at $ T_x ( partial M) $ (one towards the inside and one towards the outside). I am 100% sure that it is only a trivial linear algebra.

Because if the remaining space has a dimension $ 1 $, so it's a linear subspace. But how do we know that the rest would be orthogonal?