Making sense of the quaternion product

Question

I'm learning quaternions $\,Q=a+bi+cj+dk\,$ with $4$ real components $a,b,c,d\in\mathbb{R}$. I'm familiar with the scalar-vector representation of $\,Q=a+\vec{v}$, where $\,\vec{v}=bi+cj+dk\in\mathbb{R}^3$. But I'm not so familiar with the double-complex representation of $\,Q=(a+bi)\!+\!(c+di)j\,$ using $\,ij=k\,$ for the quaternions $\mathbb{H}$. I'm trying to make sense of the product of two quaternions

$$(z_1+w_1j)(z_2+w_2j)=(z_1z_2-w_1w_2^*)+(z_1w_2+w_1z_2^*)j.$$

I understand how this is derived. But I'm looking for a geometric interpretation of its meaning. In the scalar-vector representation, I could understand

$$(a_1+\vec{v}_1)(a_2+\vec{v}_2)=(a_1a_2-\vec{v}_1\cdot\vec{v}_2)+(a_2\vec{v}_1+a_1\vec{v}_2+\vec{v}_1\times\vec{v}_2),$$

by decomposing $\vec{v}_2=\vec{v}_{2\parallel}+\vec{v}_{2\perp}$ into components along $\vec{v}_1$ and perpendicular to $\vec{v}_1$. Then the first part $a_2+\vec{v}_{2\parallel}$ satisfies the complex product and the second part $\vec{v}_{2\perp}$ gets rotated about $\vec{v}_1$ following the right-hand rule. The geometry becomes clearer if $\,a_1+\vec{v}_1=r_1(\cos\theta_1+\hat{v}_1\sin\theta_1)\,$ is factorized into a norm times a unit quaternion. I'm not so sure what is the right way to understand the double-complex representation. Please help me. Thanks!

Answer 1

I found an answer in this question: Quaternions as an Algebra. It is difficult to understand the quaternion product by viewing $\,Q=(z,w)\,$ as an ordered pair of complex numbers. The mapping

$$Q=(z,w)\leftrightarrow\begin{pmatrix} z & w\\ -w^* & z^* \end{pmatrix}\in M_{2\times 2}(\mathbb{C})$$

makes the product easy to understand. We have

$$\begin{pmatrix} z_1 & w_1\\ -w^*_1 & z^*_1 \end{pmatrix} \begin{pmatrix} z_2 & w_2\\ -w^*_2 & z^*_2 \end{pmatrix}=\begin{pmatrix} z_1z_2-w_1w_2^* & z_1w_2+w_1z_2^*\\ -w_1^*z_2-z_1^*w_2^* & -w_1^*w_2+z_1^*z_2^* \end{pmatrix}.$$ The two complex numbers $\,z_1z_2-w_1w_2^*\,$ and $\,z_1w_2+w_1z_2^*\,$ are reproduced and the structure of the matrix is preserved. When $|z|^2+|w|^2=1$, the matrix corresponding to $Q=(z,w)$ is $\mathrm{SU}(2)$, which therefore has the geometric interpretation of unitary rotations. The norm of the quaternion $|Q|=(|z|^2+|w|^2)^{1/2}$ amplifies (or minifies) a $2\times 1$ complex vector while the $\mathrm{SU}(2)$ part rotates it. Actually the same construction applies to complex numbers $z=(a,b)$. We have

$$z=(a,b)\leftrightarrow\begin{pmatrix} a & b\\ -b & a \end{pmatrix}\in M_{2\times 2}(\mathbb{R}).$$

When $\,a^2+b^2=1$, the matrix corresponding to $\,z\,$ is $\,\mathrm{SO}(2)$. The difference between $\mathrm{SO}(2)$ and $\mathrm{SU}(2)$ is that $\mathrm{SO}(2)$ is commutative, but $\mathrm{SU}(2)$ is not.

Answer 2

Here is a geometric description of what happens. I won't bother to show that the math gives you this result, it will be obvious when you look at it.

First off, let's ignore everything but unit quaternions. If you have two quaternions Q and R, |QR|=|Q||R|. There's nothing special about that. Anything that isn't obvious will happen with unit quaternions too.

You can always describe a quaternion in terms of two separate complex numbers. For example, if you have Q=[t,x,y,z] then one way to split it up is into [t,z] and [x,y]. Two separate complex planes.

Q shows you a way to split things. t and the (x,y,z) vector give one plane A. And in 3 dimensions, if |(x,y,z)| > 0 there is a plane perpendicular to (x,y,z). That's the second plane B. Any quaternion number can be split up into two complex numbers, one in plane A and one in plane B.

Here's what happens to the geometry when you have a quaternion R and you do QR. Split R into two complex numbers, Ra in plane A and Rb in plane B. You rotate Ra by the angle that has t as its cosine and |(x,y,z)| as its sine. You rotate Rb by the same angle.

Here's what happens when you do RQ. You rotate Ra by the same angle as before, and you rotate Rb the opposite direction.

That's all there is to it. You can think of quaternion multiplication as two simultaneous complex rotations.

If you do QRQ then the two rotations in the A plane add, and the two rotations in the B plane cancel.

If you instead find Qinverse and multiply on the right by that, then the two rotations in the B plane add, and the rotations in A cancel.

It's simple and easy to visualize.

Is there anything more to it than that?

Well, there's this. The difference between [t,x,y,z] and [t,-x,-y,-z] is that one way you wind up looking at your B plane upside down. There's nothing special about quaternions in that, those two ways to look at it are inherent in the geometry.