Editing Dimensional analysis (section)

==== Directed dimensions ====
As an example of the usefulness of the first approach, suppose we wish to calculate the [[trajectory#Range and height|distance a cannonball travels]] when fired with a vertical velocity component <math>v_\text{y}</math> and a horizontal velocity component {{tmath|v_\text{x} }}, assuming it is fired on a flat surface. Assuming no use of directed lengths, the quantities of interest are then {{mvar|R}}, the distance travelled, with dimension L, {{tmath|v_\text{x} }}, {{tmath|v_\text{y} }}, both dimensioned as T<sup>−1</sup>L, and {{mvar|g}} the downward acceleration of gravity, with dimension T<sup>−2</sup>L.

With these four quantities, we may conclude that the equation for the range {{mvar|R}} may be written:
: <math>R \propto v_\text{x}^a\,v_\text{y}^b\,g^c .</math>

Or dimensionally
: <math>\mathsf{L} = \left(\mathsf{T}^{-1}\mathsf{L}\right)^{a+b} \left(\mathsf{T}^{-2}\mathsf{L}\right)^c</math>
from which we may deduce that <math>a + b + c = 1</math> and {{tmath|1=a + b + 2c = 0}}, which leaves one exponent undetermined. This is to be expected since we have two fundamental dimensions T and L, and four parameters, with one equation.

However, if we use directed length dimensions, then <math>v_\mathrm{x}</math> will be dimensioned as T<sup>−1</sup>L<sub>{{math|x}}</sub>, <math>v_\mathrm{y}</math> as T<sup>−1</sup>L<sub>{{math|y}}</sub>, {{mvar|R}} as L<sub>{{math|x}}</sub> and {{mvar|g}} as T<sup>−2</sup>L<sub>{{math|y}}</sub>. The dimensional equation becomes:
: <math>
  \mathsf{L}_\mathrm{x} =
    \left({\mathsf{T}^{-1}}{\mathsf{L}_\mathrm{x}}\right)^a
    \left({\mathsf{T}^{-1}}{\mathsf{L}_\mathrm{y}}\right)^b 
    \left({\mathsf{T}^{-2}}{\mathsf{L}_\mathrm{y}}\right)^c
</math>
and we may solve completely as {{math|1=''a'' = 1}}, {{math|1=''b'' = 1}} and {{math|1=''c'' = −1}}. The increase in deductive power gained by the use of directed length dimensions is apparent.

Huntley's concept of directed length dimensions however has some serious limitations:
* It does not deal well with vector equations involving the ''[[cross product]]'',
* nor does it handle well the use of ''angles'' as physical variables.

It also is often quite difficult to assign the L, L<sub>{{math|x}}</sub>, L<sub>{{math|y}}</sub>, L<sub>{{math|z}}</sub>, symbols to the physical variables involved in the problem of interest. He invokes a procedure that involves the "symmetry" of the physical problem. This is often very difficult to apply reliably: It is unclear as to what parts of the problem that the notion of "symmetry" is being invoked. Is it the symmetry of the physical body that forces are acting upon, or to the points, lines or areas at which forces are being applied? What if more than one body is involved with different symmetries?

Consider the spherical bubble attached to a cylindrical tube, where one wants the flow rate of air as a function of the pressure difference in the two parts. What are the Huntley extended dimensions of the viscosity of the air contained in the connected parts? What are the extended dimensions of the pressure of the two parts? Are they the same or different? These difficulties are responsible for the limited application of Huntley's directed length dimensions to real problems.