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When measurable quantities are raised to an integer power, the same is done to the dimensional symbols attached to those quantities; this corresponds to scalar multiplication in the module. A basis for such a module of dimensional symbols is called a set of base quantities , and all other vectors are called derived units.
The first table lists the fundamental quantities used in the International System of Units to define the physical dimension of physical quantities for dimensional analysis. The second table lists the derived physical quantities. Derived quantities can be expressed in terms of the base quantities.
Quantities having dimension one, dimensionless quantities, regularly occur in sciences, and are formally treated within the field of dimensional analysis.In the 19th century, French mathematician Joseph Fourier and Scottish physicist James Clerk Maxwell led significant developments in the modern concepts of dimension and unit.
As said in the introduction, dimensional homogeneity is the quality of an equation having quantities of same units on both sides. A valid equation in physics must be homogeneous, since equality cannot apply between quantities of different nature. This can be used to spot errors in formula or calculations.
A systems of quantities relates physical quantities, and due to this dependence, a limited number of quantities can serve as a basis in terms of which the dimensions of all the remaining quantities of the system can be defined. A set of mutually independent quantities may be chosen by convention to act as such a set, and are called base quantities.
A free vector is a vector quantity having an undefined support or region of application; it can be freely translated with no consequences; a displacement vector is a prototypical example of free vector. Aside from the notion of units and support, physical vector quantities may also differ from Euclidean vectors in terms of metric.
Historically, vectors were introduced in geometry and physics (typically in mechanics) for quantities that have both a magnitude and a direction, such as displacements, forces and velocity. Such quantities are represented by geometric vectors in the same way as distances, masses and time are represented by real numbers.
While natural unit systems simplify the form of each equation, it is still necessary to keep track of the non-collapsed dimensions of each quantity or expression in order to reinsert physical constants (such dimensions uniquely determine the full formula). Dimensional analysis in the collapsed system is uninformative as most quantities have the ...