physical vector

physical vector

Definition.

Let

ℒ

be a

collection

of labels

α

∈

ℒ

. For each

ordered
pair

of labels

(

α

,

β

)

∈

ℒ

×

ℒ

let

ℳ

α

β

be a

non-singular

n

×

n

matrix, the
collection of all such satisfying the following functor-like

consistency

conditions:

•

For all

α

∈

ℒ

, the matrix

ℳ

α

α

is
the

identity matrix

.

•

For all

α

,

β

,

γ

∈

ℒ

we have

ℳ

α

γ

=

ℳ

β

γ

⁢

ℳ

α

β

,

where the

product

in
the right-hand side is just ordinary

matrix multiplication

.

We then impose an

equivalence relation

by stipulating that for all

α

,

β

∈

ℒ

and

u

∈

ℝ

n

, the pair

(

α

,

u

)

is

equivalent

to the pair

(

β

,

ℳ

α

β

⁢

u

)

. Finally,
we define a

physical vector

to be an

equivalence class

of such pairs
relative to the just-defined

relation

.

The idea behind this definition is that the

α

∈

ℒ

are labels
of various

coordinate systems

, and that the matrices

ℳ

α

β

encode the corresponding changes of

coordinates

. For label

α

∈

ℒ

and list-vector

u

∈

ℝ

n

we think of the pair

(

α

,

u

)

as the
representation of a physical vector relative to the coordinate system

α

.

Discussion.

All scientific disciplines have a need for formalization. However,
the extent to which rigour is pursued varies from one discipline to
the next. Physicists and engineers are more likely to regard
mathematics as a tool for modeling and prediction. As such they are
likely to blur the distinction between

list vectors

and physical
vectors. 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, are
called vectors. A vector is three numbers. In order to represent a
step in space,

…

, we really need three numbers, but we are
going to invent a single mathematical symbol,

𝐫

, which is
unlike any other mathematical symbols we have so far used. It is

not

a single number, it represents

three

numbers:

x

,

y

, and

z

. It means three numbers, but not only

those

three numbers, because if we were to use a different coordinate
system, the three numbers would be changed to

x

′

,

y

′

, and

z

′

.
However, we want to keep our mathematics simple and so we are going
to use the

same mark

to represent the three numbers

(

x

,

y

,

z

)

and the three numbers

(

x

′

,

y

′

,

z

′

)

. That is, we use the same mark
to represent the first set of three numbers for one coordinate
system, but the second set of three numbers if we are using the
other coordinate system. This has the advantage that when we change
the coordinate system, we do not have to change the letters of our
equations.

Surely you are joking Mr. Feynman!? What are we supposed to make of
this 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 a
specific list of numbers, but rather any of a number of possible
lists. Furthermore, the choice of which list is being used is
dependent on the

context

(choice of coordinate system), but this is
not 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 than
Feynman. The

central

weakness of his definition is that he is
unwilling to distinguish between physical vectors (quantities) and
their representation (lists of numbers). Here is an alternative
physical definition from a book by R. Aris on fluid mechanics

[

2

]

.

There are many physical quantities with which only a single
magnitude can be associated. For example, when suitable units of
mass and length have been adopted the

density

of a fluid may be
measured.

…

There are other quantities associated with a
point that have not only a magnitude but also a direction. If a
force of 1 lb weight is said to act at a certain point, we can still
ask in what direction the force acts and it is not fully specified
until this direction is given. Such a physical quantity is a

vector

.

…

We distinguish therefore between the vector as an
entity and its components which allow us to reconstruct it

in a
particular system of reference.

The set of components is
meaningless unless the system of reference is also prescribed, just
as the magnitude 62.427 is meaningless as a density until the units
are also prescribed.

…

.

Definition.

A Cartesian vector,

𝐚

, in three

dimensions

is a quantity with three components

a

1

,

a

2

,

a

3

in
the frame of reference

O

⁢

123

, which, under

rotation

of the
coordinate frame to

O

⁢

1

¯

⁢

2

¯

⁢

3

¯

, become components

a

¯

1

,

a

¯

2

,

a

¯

3

, where

a

¯

j

=

l

i

⁢

j

⁢

a

i

.

The vector

𝐚

is to be regarded as an entity, just as the
physical quantity it represents is an entity. It is sometimes
convenient to use the bold face

𝐚

to show this. In any
particular coordinate system it has components

a

1

,

a

2

,

a

3

, and
it is at other times convenient to use the typical component

a

i

.

Here we see a carefully drawn distinction between physical quantities
and the numerical measurements that represent them. A system of
measurement, i.e. a choice of units and or coordinate axes, turns
physical quantities into numbers. However the correspondence is not
fixed, but varies according to the choice of measurement system. This
point of view can be formalized by representing physical vectors as
labeled list vectors, the label specifying a choice of measurement
systems. The actual vector is then defined to be an equivalence class
of such labeled list vectors.

References

1

R. Feynman, R. Leighton, and M. Sands, “Lectures on
Physics”,
11-4, Vol. I, Addison-Wesley.

2

R. Aris, “Vectors, Tensors and the Basic Equations of
Fluid Mechanics”, Dover.

Title

physical vector

Canonical name

PhysicalVector

Date of creation

2013-03-22 12:51:53

Last modified on

2013-03-22 12:51:53

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

5

Author

rmilson (146)

Entry type

Definition

Classification

msc 15A03

Classification

msc 15A90

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