Re: i,j,k things.

[Ken Caviness <caviness@SOUTHERN.EDU>]

Je 02:53 AM 9/21/99 -0400, John Denker skribis:

> 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.
...

Right.  Last time I sat down to make that point I first read the remaining
messages in the thread and found that others had said it already and, no
doubt, better.  But the thread continues -- much to my surprise, so I'll
jump in after all.  So a couple of points:

1.  unit vector = vector with unit length (i.e., length = 1, in the units
being used)

2.  We most often use the basis {i,j,k}, a set of orthogonal unit vectors,
related by the right hand rule (i.e., i x j = k), but frequently other
bases are more convenient, hence spherical-polar coordinates, etc., etc.

3.  In intro classes we define a vector as something which "has magnitude
and direction".  This is equivalent to saying that a vector can be
decomposed into components.  Then every vector can be represented in terms
of a basis set by an ordered n-tuple, in 3 dimensions using cartesian
coordinates: (x,y,z).

4.  With this representation, i, j and k may be considered to be shorthand
forms for (1,0,0), (0,1,0) and (0,0,1), respectively.  This is consistent,
since for example,

        r = xi + yj + zk
          = x(1,0,0) + y(0,1,0) + z(0,0,1)
          = (x,0,0) + (0,y,0) + (0,0,z)
          = (x,y,z)

(But this works for any basis, not only for {i,j,k}.)

5.  Using components in 3-dimensions (and _any_ basis), a unit vector is
any vector (A,B,C) such that A^2 + B^2 + C^2 = 1.

To do more, I think we may need linear algebra.

All the best / ^Cion bonan!

Ken

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