NetNews Usenet Archive 1993 #3/NN_1993_3.iso/spool/sci/math/18794

Internet Message Format | 1993-01-28 | 1.9 KB

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From: [e-mail] (Matt McIrvin)
Newsgroups: sci.math,sci.physics
Subject: Re: Vectors Question
Message-ID: <[e-mail]>
Date: 26 Jan 93 20:03:56 GMT
Article-I.D.: husc.mcirvin.728078636
References: <[e-mail]>
Lines: 28
Nntp-Posting-Host: husc8.harvard.edu

<[e-mail]> writes:

> Let us set c = a cross b. In my mind, if a and b are both polar or they are
> both axial then c is axial. I do this by picturing reflections performed on
> the vectors, and I am fairly certain of this result. If we try to cross two
> vectors of different types, then the c vectors before and after the
> reflection (c and c') differ in magnitude as well as direction and so
> I feel that it's impossible to cross different type vectors. Be that as it
> it may you won't get a polar vector by crossing a polar with and axial
> (Unless my mental pictures are totally wrong.)

I think you will, actually.  The terms axial and polar don't refer to
mirror reflections, exactly; they refer to how the vector transforms under
the transformation x -> -x, where x is a position vector.  In general
that's equivalent to a mirror reflection combined with some rotation; you
can think of it as mirror reflection through *all three* coordinate planes.
Under this space inversion a polar vector reverses direction and an
axial vector doesn't.

Now suppose a is axial and b is polar.  The product a x b points in
a direction given by the right hand rule, of course, and its magnitude
is given by |a||b| sin theta where theta is the angle between the two
vectors.  a is unchanged and b goes to -b.  This leaves theta unchanged,
as well as |a| and |b|; all that changes is the direction given by the
right-hand rule.  Therefore the product reverses direction and is a
polar vector.
--
Matt McIrvin