Re: unit vector is dimensionless, how to draw when coordinates for length?


i.love.jeevitha@xxxxxxxxx wrote:
Randy Poe wrote:
i.love.jeevitha@xxxxxxxxx wrote:
Virgil wrote:
A "unit vector" is of length 1 unit using whatever are the current units
of distance.


Sorry I realized that I probably should have asked this question in
sci.physics and later started a thread there. As for your answer, if
that was true I would be very happy.

It's true. Virgil is correct.

Unfortunately the formula for the
unit vector takes out the effect of any unit.

unit vector = v / ||v||

Let's just see what the resulting vector is, shall we?

Suppose v is a vector of length 3 meters in some direction.
Then ||v|| = 3, in meters. So u = v/||v|| is a vector 1/3 the
length of v, in the same direction as v.

That is, it is a vector 1 m long.

Now suppose v is measured in centimeters. Then ||v|| = 30.
So u = v/||v|| is a vector 1/30 the length of v, in the same
direction as v.

That is, it is a vector 1 cm long.

Note that the "unit vector" isn't unique. The second unit
vector is obviously a lot shorter than the first unit vector.


Hmmm. In v / ||v|| seems like the units are cancelling out. Are you
sure? I'm getting diametrically opposite answers from every different
person I ask. :(

There may be two different conventions going on here.

Ultimately it depends on what you want to do with it. If it
is a vector, it has a magnitude. The idea of a vector that
has direction but not magnitude is a little hard for me to
swallow.

UNLESS.... you don't think of it as a vector, but as a thing
which encapsulates the direction information of a vector.

In that view (which is probably closer to what others are
telling you), the unit "vector" (which has no units and is
not a vector) would always appear multiplied by a thing
with units.

So my 1 cm vector would be 1 cm * u.
My 1 m vector would be 1 m * u.

I can see the self-consistency in this system, but unfortunately
it leads to the confusing fact that the "unit vector" is not
the same as a "vector of length 1 unit".

- Randy

.

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