Expressing common vector operations in terms of quaternions

Question

Since everything we do with vectors could also in theory be expressed using quaternions (or octonions & higher-order -nions), is it possible to have common vector operations like the cross product or the dot product expressed in terms of quaternions? How are they called and what would they look like?

Answer 1

If you express* two vectors $v_1,v_2$ from $\mathbb R^3$ as quaternions $q_1,q_2$ with real part $0$, then the real part of $q_1q_2$ is $-v_1\cdot v_2$ and the pure quaternions part is $v_1\times v_2$.

This follows from the formula $$ (r_1,\ \vec{v}_1) (r_2,\ \vec{v}_2) = (r_1 r_2 - \vec{v}_1\cdot\vec{v}_2,\ r_1\vec{v}_2+r_2\vec{v}_1 + \vec{v}_1\times\vec{v}_2) $$ as given for example here.

So they are both encoded: $v_1\cdot v_2=-\text{RealPart}(q_1q_2)$

$v_1\times v_2=\text{PureQuaternionPart}(q_1q_2)$

where the real part of $a+bi+cj+dk$ is $a$ and the pure quaternion part is $(b,c,d)$.

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(*) Concretely, if $v=(b,c,d)$, the quaternion representing it is $bi+cj+dk$.

Answer 2

Well, at first glance quaternions do not usually represent vectors, but unit quaternions represent rotation matrices, with multiplication of quaternions being something like a matrix product. Quaternions form an algebra, i.e. the important operations one can do with them are addition, multiplication with real numbers, and multiplication of quaternions.

Of course, they form a 4d vector space, so anything you can do with those you can also do here. There are generalizations of cross and dot product to any dimension (scalar product and wedge product), but they do not depend on the extra structure, i.e. the multiplication, quaternions have. Therefore these probably aren't what you are interested in.

Also, quaternions form a division ring, so that one can define left and right vector spaces over quaternions. One can also define scalar and wedge products on those, and I do think that would give an interesting theory, since there is much more structure here than on modules over arbitrary commutative rings. Maybe the most interesting way for you to go would be this direction, i.e. learn a bit about Euklidean and unitary vector spaces, tensor products, exterior algebras and how this can be generalized to modules over rings.