Help w/pseudovectors
Wan Li Zhu
Hi, I'm really stuck on a physics problem regarding pseudovectors
and I'd appreciate any help:
a) Is the cross product of 2 pseudovectors a vector, or a pseudovector?
How about the cross product of a vector and a pseudovector? Is the
vector triple product of 3 bectors a vector, or a pseudovector? Name 2
pseudovector quantities in classical mechanics.
b) How does the scalar triple product of 3 vectors transform under
inversion? (such an object is called a pseudoscalar).
[Moderator's note: as usual, questions which may be seeking answers
for help with homework are best dealt with, not by simply giving the
answer, but by helping the questioner figure out the answers himself.
- jb]
--
Wan Li Zhu
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Toby Bartels
Wan Li Zhu <zh...@okcom.net> wrote:
>a) Is the cross product of 2 pseudovectors a vector, or a pseudovector?
>How about the cross product of a vector and a pseudovector? Is the
>vector triple product of 3 bectors a vector, or a pseudovector? Name 2
>pseudovector quantities in classical mechanics.
When you change the sign of all the vectors,
a pseudovector does not change sign.
You can see this in the cross product A = b x c,
where I am writing pseudovectors in uppercase and vectors in lowercase.
If b changes to -b and c changes to -c,
A changes to -b x -c = -(-b x c) = + b x c = A.
The two minus signs cancel each other.
That's how we know that the cross product of two vectors is a pseudovector.
Now look at ? = B x C. Is ? the vector a or the pseudovector A?
Well, see if it changes sign when all the vectors change sign.
What does B change to? (Remember it's a pseudovector.)
What does C change to? Now multiply them, and see what happened to a or A.
To find examples of pseudovectors in classical mechanics,
I suggest that you look to the mechanics of rotations,
because that subject is rife with pseudovectors.
The basic quantities of linear mechanics, displacement, momentum, and force,
are all vectors. So find something which is the cross product of two of these.
That quantity will be a pseudovector.
>b) How does the scalar triple product of 3 vectors transform under
>inversion? (such an object is called a pseudoscalar).
Now you have a.(b x c).
What happens if you change a to -a, b to -b, and c to -c?
Multiply them together and see what happens to a.(b x c).
-- Toby
to...@ugcs.caltech.edu