Buckingham pi theorem: alternative pi terms and orthogonality A question on the buckingham pi theorem: It provides one with the socalled pi terms forming linearly independent quantities based on the relevant dimensions occuring in the problem. They are a particular set of basis vectors spanning the kernel.
The vector addition operation is multiplying the dimensions and the scalar multiplication operation is raising to a power.
It is frequently stated that these pi terms are not unique, but as far as I've seen alternative examples are never given.
Am I correct in saying that alternatives are simply other linear combinations of the initially determined pi-terms, nothing more?
Furthermore: is there something such as orthogonality in this vector space, because for that to happen there should be some inner product right? Is there such an inner product and would a gram-schmidt process generate alternative (orthogonal) pi term sets?
 A: The $\pi$'s are not always unique because there may be more than one solution to the linear system.  Obviously, if $\pi_1$ and $\pi_2$ are dimensionless, so is their product $\pi_1\times \pi_2$, and their product will correspond to a linear combination of the original solutions to the system.  There are plenty of examples in hydrodynamics where more than one dimensionless variable can be constructed from the setup of the problem.
Mathematically there is no way of selecting one solution over another, but physically there might be some variables that are more easily accessible to experiment than others, there might be others which are not accessible at all, or some that remain constant.  This is usually what drives the choice of one solution over another.
Note that one tends to look for solutions where the coefficients are "simple" rational functions, i.e. all integers or maybe some half-integers.  Hence Gram-Schmidt or orthonormality is unlikely to bring much benefit in terms of understanding the scaling behaviour of the various $\pi$'s.
