The definition of a *vector* as an entity with both magnitude
and direction can be generalized if we realize that "direction"
can be defined in more dimensions than the usual 3 spatial
directions, "up-down, left-right, and back-forth," or even
other dimensions *excluding* these three. The more
general definition would read,

DEFINITION:avectorquantity is one which hasseveral independent attributeswhich areall measured in the same unitsso that "transformations" are possible. (This last feature is only essential when we want the advantages of mathematical manipulation; it is not necessary for the concept of multi-dimensional entities.)

We can best illustrate this generalization with an example of a vector that has nothing to do with 3-D space:

EXAMPLE: theCost of Living, , is in a sense a true vector quantity (although the Cost of Livingindexmay be properly thought of as ascalar, as we can show later).

To construct a simple version, the Cost of Living can be taken to include:

*C*_{1}=**housing**(e.g., monthly rent);*C*_{2}=**food**(e.g., cost of a quart of milk);*C*_{3}=**medical**service (e.g., cost of a bottle of aspirin);*C*_{4}=**entertainment**(e.g., cost of a movie ticket);*C*_{5}=**transportation**(e.g., bus fare);-
etc.
(a finite number of "
*components*.")

Thus we can write
as an ordered sequence of numbers representing
the values of its respective "components":

We would normally go on until we had a reasonably "complete"
list - i.e., one with which the cost of any additional
item we might imagine could be expressed *in terms of*
the ones we have already defined.
The technical mathematical term for this condition is that
we have a "*complete basis set*"
of components of the Cost of Living.
Now, we can immediately see an "inefficiency" in the way
can been "composed:"
As recently as 1975, it was estimated to take approximately
one pound of gasoline to grow one pound of food in the U.S.A.;
therefore the cost of **food** and the cost of **transportation**
are obviously *not independent*! Both are closely tied to
the cost of **oil**. In fact, a large number of the components
of the cost of living we observe are intimately connected
to the cost of oil (among other things). On the other hand
(before we jump to the fashionable conclusion that these
two components should be replaced by oil prices alone),
there is *some* measure of independence in the two
components. How do we deal with this quantitatively?
To reiterate the question more formally, how do we
quantitatively describe the extent to which certain components
of a vector are superfluous (in the sense that they merely
represent combinations of the other components) *vs.* the extent
to which they are truly "independent?" To answer, it is
convenient to revert to our old standby, the (graphable)
analogy of the **distance** vector in *two dimensions*.

Suppose we wanted to describe the position of any
point *P* in the "*x*-*y* plane." We could draw the two axes
"*a*" and "*b*" shown above.
The position of an arbitrary point *P*
is uniquely determined by its (*a*,*b*) coordinates,
defined by the prescription that
to change *a* we move parallel to the *a*-axis and to change *b*
we move parallel to the *b*-axis. This is a unique and quite
legitimate way of specifying the position of any point
(in fact it is often used in crystallography where the
orientation of certain crystal axes is determined by nature);
yet there is something vaguely troubling about this choice of
coordinate axes. What is it?
Well, we have an intuitive sense of "up-down" and "sideways"
as being *perpendicular*, so that if something moves "up"
(as we normally think of it), in the above description
the values of both *a* and *b* will change. But isn't our
intuition just the result of a well-entrenched convention?
If we got used to thinking of "up" as being in the "*b*"
direction shown, wouldn't this cognitive dissonance dissolve?
No. In the first place, nature provides us with an
unambiguous characterization of "down:" it is the direction
in which things fall when released; the direction a string
points when tied to a plumb bob. "Sideways," similarly,
is the direction defined by the surface of an undisturbed
liquid (as long as we neglect the curvature of the Earth's
surface). That is, gravity fixes our notions of "appropriate"
geometry. But is this in turn arbitrary (on nature's part)
or is there some good reason why "independent" components
of a vector should be perpendicular? And what exactly do
we *mean* by "perpendicular," anyway? Can we define the concept
in a way which might allow us to generalize it to other
kinds of vectors besides space vectors?

The answer is bound up in the way Euclid found to express
the geometrical properties of the world we live in; in particular,
the "metric" of space - the way we define
the **magnitude** (*length*) of a vector.
Suppose you take a ruler and turn it at many angles;
your idea of the *length* of the ruler is *independent*
of its *orientation*, right?
Suppose you used the ruler to make off
distances along two perpendicular axes, stating that these were
the horizontal and vertical components (*x*,*y*) of a distance vector.
Then you use the usual "parallelogram rule" to locate the tip
of the vector, draw in a line from the origin to that point,
and put an arrowhead on the line to indicate that you have a
vector. Call it
"". *You can use the same ruler, held at
an angle, to measure the length **r** of the vector. * Pythagoras
gave us a formula for this length. It is

This formula is the key to Euclidean geometry, and is the working definition of perpendicular axes:

("Orthogonal" and "normal" are just synonyms for "perpendicular.") We could call the "Cost of Living Index" if we liked. There is a problem now. Our intuitive notion of "independent" components is tied up with the idea that one component can change without affecting another; yet as soon as we attempt to be specific about it, we find that we cannot even define a criterion for formal and exact independence (orthogonality) without generating a new notion: the idea of a magnitude as defined by Eq. (3). Does this definition agree with out intuition, the way the "ruler" analogy did? Most probably we

Jess H. Brewer - Last modified: Fri Nov 13 17:28:12 PST 2015