Re: i,j,k things.

[John Denker <jsd@MONMOUTH.COM>]

At 10:29 AM 9/22/99 -0400, Ludwik Kowalski wrote:
> We call i,j,k unit vectors because
> they become vectors, not because they are vectors by themselves.

I wouldn't have said that.  If i, j, or k is a unit vector, then it is a
vector --- as surely as a gallon jug is a jug.  Any vector that happens to
have unit length is a unit vector.  Any jug that happens to hold one gallon
is a gallon jug.  There's nothing deep about the concept.

> A pure number means nothing
> in practice unless we say what it represents.

That's a bit overstated (and likely to unnecessarily inflame mathematicians)
but in the context of the following sentence I know what you mean.

> In one problem it may
> be a number of apples, in another a number of volts, or the length
> of an arrow.

True enough.

> In the same way i,j,k mean nothing unless a frame of
> reference is chosen (or implied) and a physical unit is chosen to
> represent a physical quantity. We call i,j,k unit vectors because
> they become vectors, not because they are vectors by themselves.

I wouldn't say that at all.  Indeed the shoe is on the other foot:  we can
arbitrarily choose three orthogonal directions, and arbitrarily choose units
of measurement (e.g. the proverbial furlongs per fortnight) -- and thereby
construct unit vectors which we might call i, j, and k.  Having done this,
we can use {i,j,k} as a frame of reference -- or not.  The vectors i, j,
and k remain vectors whether or not we use them as a basis.

The laws of physics don't care about units of measurement, and don't care
about the choice of basis.  The choices are arbitrary.  The only real
requirement is that when communicating your results, you must be clear about
what choices you've made.

There are some calculations that are easier to do component-by-component.
In such cases, we can resolve a vector into components any time we want.
Resolving into components is tantamount to multiplying by the identity
operator (i.e. the identity operator represented as a sum of projection
operators).  We can always do that if we want.  There are innumerably many
different ways to do that.  But we don't have to use any particular
representation of the identity operator unless we want to.

> In one problem 1i (written as i) will stand for the horizontal
> velocity (in m/s), in another for the force pulling an object along
> an inclined plane, etc.

True.

> This is not related to the choice of a scale
> (for example one inch on paper for 0.25 N, 1 cm for 5 m/s or
> 1 mm for 1 a.u.). Scales become important only when problems
> are solved graphically rather than algebraically.

Also true.

______________________________________________________________
John S. Denker                                jsd@monmouth.com