Quaternions, Rotations and Real numbers

I haven't really formally studied Algebra at anywhere near this level, but I was told about the existence of Quaternions a few years ago and I find them really cool. I also like how pure quaternions are analogous to cross products in $R^3$, and that it gives me a way of doing it algebraically rather than relying on hand rules.

Over the course of using them it's my understanding that multiplying an element by another, both imaginary numbers, rotates it in some manner so as to be orthogonal to both, similar to rotations of $\frac{\pi}{2}$ in the space with one real and one imaginary axis when multiplied by $i$, which is why it's useful for cross products as it will rotate it to be orthogonal to a plane spanned by linear combinations of those two vectors [afaik].

What I can't intuit though is why for any pure quaternion multiplication, there's a negative real part [the scalar product] if it's meant to correspond to a rotation within $R^3$ and the space of pure Quaternions is Isomorphic to $R^3$. Similarly I don't understand why $i^2 = j^2 = k^2 = $ A negative real.

Solution 1:

Don't try interpret quaternion multiplication $q_1 q_2$ as immediatedly connected to rotations; it's more indirect.

The operation which performs a rotation is $q u q^*$, where $u$ is a pure vector quaternion (to be rotated), and $q$ is a unit quaternion $$ q = \cos(\alpha/2) + \sin(\alpha/2) \, n , $$ where $n$ is a unit vector giving the axis of rotation, and $\alpha$ is the angle of rotation. (And $q^* = \cos(\alpha/2) - \sin(\alpha/2) \, n$ is the conjugate.)

Then a general quaternion (a constant times a unit quaternion) can be thought of as representing a linear transformation which is a combination of a rotation and a scaling, and the multiplication of two quaternions is an algebraic way of computing the composition of two such linear transformations: $$ q_1 \bigl( q_2 u q_2^* \bigr) q_1^* = (q_1 q_2) u (q_1 q_2)^* . $$ For example, a pure vector (unit) quaternion performs a 180 degree rotation around the corresponding axis, but when you compose two 180 degree rotations you don't get a new 180 degree rotation, unless the two axes of rotation happen to be orthogonal. So it makes perfect sense to have a scalar part (nonzero in general) in the quaternion product; it is related to the angle of the composed rotation.