Vectorial issue of friction

[schrodingerwitch]

Considering an stopped object in a horizontal plane, the frictional force between the object and the plane would be the product of the friction coefficient (static or kinetic if there was movement between the surfaces) by normal. Since the normal in this case would be given by N (vector) = - mg j (unit), we would have the frictional force given by F (vector) = - mgμ j (unit).
But we know that the frictional force must act against the direction of the object's movement, so its unit vector must have direction î (unitary).

Is this change in the unit vector simply a convention that comes from agreement with experimental physics or is there some kind of vector transformation that makes this vector j^ already become a vector î?

The books I researched were Halliday, Tipler, Moysés and Young and Freedman. In none of them did I see comments about it, so I think it might be trivial, but I wanted to clear that doubt. The only book that talked about it was Alonso and Finn - A University Course, but it was just a comment and I found it confusing.

 

[PeroK]

Is this change in the unit vector simply a convention that comes from agreement with experimental physics or is there some kind of vector transformation that makes this vector j^ already become a vector î?

It's a consequence of the law of friction, where the resisting force of friction is in the direction of relative motion of the surfaces and depends on the magnitude of the normal force between the surfaces. The unit vectors are implied by and deduced from the law.

 

[Doc Al]

I'll restate what @PeroK just explained, but slightly differently.

Considering an stopped object in a horizontal plane, the frictional force between the object and the plane would be the product of the friction coefficient (static or kinetic if there was movement between the surfaces) by normal. Since the normal in this case would be given by N (vector) = - mg j (unit), we would have the frictional force given by F (vector) = - mgμ j (unit).

No, the frictional force is not the product of the friction coefficient (a scalar) and the normal force (a vector) acting on the surface. (If so, the friction force would be perpendicular to the surface, as you note.) Instead, the magnitude of the friction force (for kinetic friction, at least) is given by the product of the friction coefficient (a scalar) and the magnitude of the normal force (another scalar).

The direction of the friction force (as @PeroK explained) always opposes the sliding between the surfaces and is parallel to those surfaces.