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Re: zero-length vectors (was: displacement)



At 09:46 AM 1/10/00 -0500, Peter Schoch wrote:

Now, I have asked my math colleages about this, and they say that 0
magnitude with 0 degrees makes perfect mathematical sense to them.

Perfect?

That answer exhibits remarkably little mathematical sophistication.

So.... I'm happy enough to leave it at that -- unless someone else has a
better answer?

Before we get to "better" answers -- how about some equally-good
answers: what about
-- 0 magnitude with 13.97 degrees?
-- 0 magnitude with 77.43 degrees?

===========

Seriously, given a vector with components X and Y, we are considering the
function
theta = arg(X, Y)

Now this function is well defined everywhere in the plane except for one
point at the origin. Alas it is not defined at the origin, and there's
nothing you can do about it. There is, for instance, no analytic
continuation. The limit(as X and Y go to zero) does not exist. You might
wish that the limit exists. You might wish that the limit is equal to
zero. But it does not exist.

The nonexistence of this limit is related to the fact that arg(,) cannot be
a continuous function (even if we stay away from the origin).

The original question was asked by a high school student who may be in the
unfortunate position of knowing enough about vectors to ask the question
but not enough about limits and continuity to understand a detailed
answer. In such a case, the following may be of some help:
-- Consider the analogy to the expression 0/0 -- what is the value of
0/0?? Is zero the answer? No, it certainly does not make "perfect
mathematical sense" to say zero is the answer.
-- Consider the analogy to the question "what is three miles north of the
north pole?" ... there are some questions that should be left unanswered.
-- For the short term, dear student, take my word for it, there is *no*
number that properly represents the arg() of a zero-length vector. In the
longer term, in the opening days of your first calculus course you will
learn hundreds of examples of this ilk, and will learn the tools for
thinking about them properly.