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: a vector quantity is one which has several independent attributes which are all measured in the same units so 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: the Cost of Living, , is in a sense a true vector quantity (although the Cost of Living index may be properly thought of as a scalar, as we can show later).
To construct a simple version, the Cost of Living can be taken to include:
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