I started from wandering if the cross product (a product between two vectors that gives a vector) can be abstracted like dot product (a product between two vectors that gives a scalar) is abstracted in abstract inner spaces in linear algebra.
In abstract algebra (Gallian) a vector space V over a field F is defined as an Abelian group, so it seems that it has to inherit group internal and external products. Groups internal and external products are operations between groups and subgroups, not between elements of them, but in abstract algebra even componentwise operations of addition and multiplication are defined for group and ring elements.
So: - in which relation are groups internal and external products with vector space products? - in which relation are group/ring componentwise operations with dot and inner products in vector spaces?