If you have two pointy arrows, and want to know if they are perpendicular, then you use the scalar product u⋅v and check if it is 0. There is no difference betw

If you have two pointy arrows, and want to know if they are perpendicular, then you use the scalar product u⋅v and check if it is 0. There is no difference between u and v in this case. You could say v is a vector, and u a function taking a vector and returning a scalar (a form, a dual vector, ...). But often that is unneccessary artifice. You can just say they are both vectors in the same vector space, which is imbued with an inner product. Declaring that some objects are forms (duals, covariant vectors, ...) is only neccessary when you want to keep track of transformation properties, for example.