On Dimensional Analysis

The dimensions of physical quantities q are interpreted as vectors q
i
(γ
i1
, γ
i2
, …,γ
in
)≡γ
i1
b
1
+γ
i2
b
2
+…+γ
in
b
n
, where the basic elements b
j
generating the vector space represent the basic quantities of the dimensional system and the coefficients γ
j
are defined by an equation. This interpretation permits the application of the theorems on vector spaces to dimensional analysis. Some results of this approach are simplified rules for the transformation of dimension and unit systems and a physically more transparent derivation of a complete set of dimensionless products by a transformation of bases. The new notation yields a sequential order of physical equations which may lead to a dimensional analysis based on appropriately selected equation groups.