Different ways of representing the group $SU(3)$ within a Geometric Algebra approach are explored. As part of this we consider characteristic multivectors for $SU(3)$, and how these are linked with decomposition of generators into commuting bivectors. The setting for this work is within a 6d Euclidean Clifford Algebra. We then go on to consider whether the fundamental forces of particle physics might arise from symmetry considerations in just the 4d geometric algebra of spacetime --- the STA. As part of this, a representation of $SU(3)$ is found wholly within the STA, involving preservation of a bivector norm. We also show how Octonions can be fully represented within the Spacetime Algebra, which we believe will be useful in making them understandable and accessible to a new community in Physics and Engineering. The two strands of the paper are drawn together in showing how preserving the octonion norm is the same as preserving the timelike part of the Dirac current of a particle. This suggests a new model for the symmetries preserved in particle physics. Following on from work by G\"unaydin and G\"ursey on the link between quarks, and octonions, and by Furey on chains of octonionic multiplications, we show how both of these fit well within our scheme, and give some wholly STA versions of the operations involved, which in the cases considered have easily understandable equivalents in terms of 4d geometry. Links with larger groups containing $SU(3)$, such as $G_2$ and $SU(8)$, are also considered.

If an initial frame of vectors $\{e_i\}$ is related to a final frame of vectors $\{f_i\}$ by, in geometric algebra (GA) terms, a {\it rotor}, or in linear algebra terms, an {\it orthogonal transformation}, we often want to find this rotor given the initial and final sets of vectors. One very common example is finding a rotor or $4\times 4$ orthogonal matrix representing rotation and translation, given knowledge of initial and transformed points. In this paper we discuss methods in the literature for recovering such rotors and then outline a GA method which generalises to cases of any signature and any dimension, and which is not restricted to orthonormal sets of vectors. The proof of this technique is both concise and elegant and uses the concept of {\it characteristic multivectors} as discussed in the book by Hestenes \& Sobczyk, which contains a treatment of linear algebra using geometric algebra. Expressing orthogonal transformations as rotors, enables us to create {\it fractional transformations} and we discuss this for some classic transforms. In real applications, our initial and/or final sets of vectors will be noisy. We show how to use the characteristic multivector method to find a ‘best fit’ rotor between these sets and compare our results with other methods.