Rotors: A practical introduction for 3D graphics (2023)

There are vectors in the wide sense, i.e. elements of a linear space. Linear spaces are a.k.a. vector spaces, where "vector" is used in the wide sense.

Then there are vectors in the strict sense, which is the sense corresponding to the etymology of the word "vector", which have additional properties over the axioms of a linear space.

Vectors in the strict sense are elements of some particular linear spaces that are associated with the translations of affine spaces, and which are also associated with geometric algebras, where the dimensions of the geometric algebras as linear spaces over the real numbers are 2^N, where N is the dimension of the set of vectors as a linear space over the real numbers.

Quaternions as a linear space over the real numbers happen to be 4-dimensional, but this 4-dimensional space has no relationship whatsoever with a 4-dimensional space that would be an extension of the familiar 3-dimensional space of the Euclidean geometry, which models the space in which we live.

Since the quaternions are means for describing transformations of the 3-dimensional space of Euclidean geometry, all applications of the quaternions include the 3D geometry in a more or less disguised form, in the same way as any application of complex numbers includes the geometry of the Euclidean plane, even if that is not obvious because the applications are described in an abstract way, using only the axioms of the field of quaternions or of the field of complex numbers.

Many applications of complex numbers in electronics or digital signal processing become far more easier to understand when one thinks about the geometric transformations of a plane that correspond to complex numbers, instead of thinking only about the axioms of the field of complex numbers. The same happens for quaternions.

The physical space in which we live and that we can imagine, is modeled mathematically as an affine space, i.e. as a space of points. We can also imagine affine spaces with more dimensions than 3.

Some linear spaces are vector spaces in the strict sense, being sets of the translations of an affine space. Other linear spaces, like the set of quaternions, are not vector spaces in the strict sense. In order to help our perception of such abstract linear spaces, we may use tools like graphs or drawings that map some part of the abstract linear space to an affine space that we can visualize, e.g. on a computer display, but we must keep in mind that this is just a mapping and that the nature of that abstract linear space is different from the spaces that we can see.