Several people have suggested teaching vectors starting
with some examples. I agree with that, but I would like
to expand the range of examples a little bit.
In particular, instead of just the question
-- what are vectors?
it would be good to also address the questions
-- why vectors? what are vectors good for?
Most(*) of the answer IMHO has to do with the
transformation properties of vectors, as illustrated
by the following example:
Build a diorama representing a small village.
1) The barbershop is one block east of the school. This is
an example of a displacement vector.
Similarly, the grocery store is two blocks northwest of
the school. This is another example of a displacement vector.
Now rotate the diorama 180 degrees. The barbershop is now
west of the school, and the store is now southeast of the
school.
However (!) the length of each separation vector is unchanged,
and the angle between any two separation vectors is unchanged.
Velocity vectors, acceleration vectors, force vectors etc.
behave similarly.
2) By way of contrast, the barbershop has a certain
temperature. When you rotate the diorama, the temperature
of the barbershop is unchanged. Temperature is not a vector.
It is just a plain old scalar.
The extension from scalars to vectors is neither the first
nor the last extension students will see. Starting with
whole numbers, they extended the concept of "how much?" to
include fractions. Later they extended it in another direction
to include negative numbers. Now they are getting to vectors.
In the future they will extend from vector space to vector field,
and from first-rank vectors to second-rank tensors etc. etc.
=====
We begin to see why vectors show up in physics. We believe
that the laws of physics should be invariant under rotation.
If you write the laws in terms of vectors, you get invariance
for free.
=====
Let's talk for a moment about what vectors are not:
Vectors are sometimes defined as objects having a magnitude
and a direction, but this has disadvantages (as well as
advantages), for several reasons. The main disadvantage is
that it defeats the main purpose, because direction is the
one thing about vectors that is not invariant under rotation.
++ the length of a vector is rotationally invariant
++ the angle between two vectors is invariant
++ you can add two vectors in a way that is invariant
++ if two vectors are equal, equality is rotationally invariant
-- the direction of a vector is not rotationally invariant
Sometimes people think that any list of three numbers is a
vector. That's just plain wrong. Counterexample: a list
of three phone numbers.
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*) Note: The "vectors" here are Euclidean vectors. This
conforms to the spirit of the original question. However,
afficianados will note that it is possible to generalize
the notion of vector to include things that don't have a
metric, and therefore don't have any notion of angle or
magnitude. For the most general definition of vector, see
e.g. http://mathworld.wolfram.com/VectorSpace.html
Imperfect examples include
-- Shopping lists. You can add 'em and multiply by scalars.
You can quibble about whether negatives are well behaved.
-- Lists of extensive thermodynamic quantities such as energy,
entropy, volume, et cetera.
This posting is the position of the writer, not that of the
Inquisition, the Gestapo, or the KGB.
This posting is the position of the writer, not that of SUNY-BSC, NAU or the AAPT.