physical vector

Definition.

Let $ℒ$ be a collection of labels $\alpha \in ℒ$. For each orderedpair of labels $(\alpha ,\beta )\in ℒ\times ℒ$ let${ℳ}_{\alpha }^{\beta }$ be a non-singular $n\times n$ matrix, thecollection of all such satisfying the following functor-likeconsistency conditions:

• For all $\alpha \in ℒ$, the matrix ${ℳ}_{\alpha }^{\alpha }$ isthe identity matrix.

• For all $\alpha ,\beta ,\gamma \in ℒ$ we have

  $${ℳ}_{\alpha }^{\gamma }={ℳ}_{\beta }^{\gamma }{ℳ}_{\alpha }^{\beta },$$

  where the product inthe right-hand side is just ordinary matrix multiplication.

We then impose an equivalence relation by stipulating that for all$\alpha ,\beta \in ℒ$ and $u\in {ℝ}^n$, the pair $(\alpha ,u)$ isequivalent to the pair $(\beta ,{ℳ}_{\alpha }^{\beta }u)$. Finally,we define a physical vector to be an equivalence class of such pairsrelative to the just-defined relation.

The idea behind this definition is that the $\alpha \in ℒ$ are labelsof various coordinate systems, and that the matrices${ℳ}_{\alpha }^{\beta }$ encode the corresponding changes ofcoordinates. For label $\alpha \in ℒ$ and list-vector$u\in {ℝ}^n$ we think of the pair $(\alpha ,u)$ as therepresentation of a physical vector relative to the coordinate system$\alpha$.

Discussion.

All scientific disciplines have a need for formalization. However,the extent to which rigour is pursued varies from one discipline tothe next. Physicists and engineers are more likely to regardmathematics as a tool for modeling and prediction. As such they arelikely to blur the distinction between list vectors and physicalvectors. Consider, for example the following excerpt from R.Feynman’s “Lectures on physics” [1]

All quantities that have a direction, like a step in space, arecalled vectors. A vector is three numbers. In order to represent astep in space, $\mathrm{\dots }$, we really need three numbers, but we aregoing to invent a single mathematical symbol, $𝐫$, which isunlike any other mathematical symbols we have so far used. It isnot a single number, it represents three numbers: $x$,$y$, and $z$. It means three numbers, but not only thosethree numbers, because if we were to use a different coordinatesystem, the three numbers would be changed to $x^{\prime }$, $y^{\prime }$, and $z^{\prime }$.However, we want to keep our mathematics simple and so we are goingto use the same mark to represent the three numbers $(x,y,z)$and the three numbers $(x^{\prime },y^{\prime },z^{\prime })$. That is, we use the same markto represent the first set of three numbers for one coordinatesystem, but the second set of three numbers if we are using theother coordinate system. This has the advantage that when we changethe coordinate system, we do not have to change the letters of ourequations.

Surely you are joking Mr. Feynman!? What are we supposed to make ofthis definition? We learn that a vector is both a physical quantity,and a list of numbers. However we also learn that it is not really aspecific list of numbers, but rather any of a number of possiblelists. Furthermore, the choice of which list is being used isdependent on the context (choice of coordinate system), but this isnot really important because we just end up using the same symbol$𝐫$ regardless.

What a muddle! Even at the informal level one can do better thanFeynman. The central weakness of his definition is that he isunwilling to distinguish between physical vectors (quantities) andtheir representation (lists of numbers). Here is an alternativephysical definition from a book by R. Aris on fluid mechanics[2].

There are many physical quantities with which only a singlemagnitude can be associated. For example, when suitable units ofmass and length have been adopted the density of a fluid may bemeasured. $\mathrm{\dots }$ There are other quantities associated with apoint that have not only a magnitude but also a direction. If aforce of 1 lb weight is said to act at a certain point, we can stillask in what direction the force acts and it is not fully specifieduntil this direction is given. Such a physical quantity is a vector. $\mathrm{\dots }$ We distinguish therefore between the vector as anentity and its components which allow us to reconstruct it in aparticular system of reference. The set of components ismeaningless unless the system of reference is also prescribed, justas the magnitude 62.427 is meaningless as a density until the unitsare also prescribed.$\mathrm{\dots }$.
Definition. A Cartesian vector, $𝐚$, in threedimensions is a quantity with three components $a_1,a_2,a_3$ inthe frame of reference $O123$, which, under rotation of thecoordinate frame to $O\overline{1}\overline{2}\overline{3}$, become components${\overline{a}}_1,{\overline{a}}_2,{\overline{a}}_3$, where

$${\overline{a}}_j=l_{ij}{a}_i.$$

The vector $𝐚$ is to be regarded as an entity, just as thephysical quantity it represents is an entity. It is sometimesconvenient to use the bold face $𝐚$ to show this. In anyparticular coordinate system it has components $a_1,a_2,a_3$, andit is at other times convenient to use the typical component $a_i$.

Here we see a carefully drawn distinction between physical quantitiesand the numerical measurements that represent them. A system ofmeasurement, i.e. a choice of units and or coordinate axes, turnsphysical quantities into numbers. However the correspondence is notfixed, but varies according to the choice of measurement system. Thispoint of view can be formalized by representing physical vectors aslabeled list vectors, the label specifying a choice of measurementsystems. The actual vector is then defined to be an equivalence classof such labeled list vectors.