Are Quaternions Just Shorthand For Normalized Vectors?

Question

The title pretty much sums this one up. I've noticed that you can create a quaternion from a normalized vector on the versor side with a 0 on the scalar side. Does this mean quaternions are just a normalized form of vectors? If not what are the differences?

Answer 1

3D vectors are an important subspace of the quaternions, and normalizing vectors and quaternions is important in applications, but that doesn't mean every quaternion is a normalized 3D vector any more than the integers being a subset of the reals means every real number is an integer. Technically, since the set of quaternions is a 4D vector space, we can and often should think of quaternions as vectors, but more often we want to think of quaternions as numbers that can be used for transformations. And while normalizing quaternions, or restricting our attention to unit quaternions, is typical and important for modelling rotations, quaternions in general do have magnitude and that's part of them being closed under addition and called an $\mathbb{R}$-algebra. Keeping this in mind is important for actually proving things about or with quaternions, and for relating quaternions to other things in abstract algebra.