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MathGroup Archive 2004

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Re: How to reprensent the result in vector form?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg51657] Re: How to reprensent the result in vector form?
  • From: "Steve Luttrell" <steve_usenet at _removemefirst_luttrell.org.uk>
  • Date: Wed, 27 Oct 2004 23:43:55 -0400 (EDT)
  • References: <cl9t1d$7m0$1@smc.vnet.net> <200410230422.AAA26310@smc.vnet.net> <clndek$o4s$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

I fully agree with you that my suggestion was not a general solution. I had 
in mind the sort of piecemeal replacement(s) you sometimes have to do at the 
end of a derivation to coax the result into the form you want. I usually 
concoct these on the fly and store them as rules.

A quick (and very dirty!) fix to the problem of matching to ALL possible 
sign combinations is

crossrule2=
  Flatten[Table[{Sign[ai3 bi2]a3_ b2_ -Sign[ai2 bi3]a2_ b3_,
          Sign[ai1 bi3]a1_ b3_ -Sign[ai3 bi1]a3_ b1_,
          Sign[ai2 bi1]a2_ b1_ -Sign[ai1 bi2]a1_ b2_}->
        myCross[{bi1 b1,bi2 b2,bi3 b3},{ai1 a1,ai2 a2,ai3 a3}],{ai1,-1,1,
        2},{ai2,-1,1,2},{ai3,-1,1,2},{bi1,-1,1,2},{bi2,-1,1,2},{bi3,-1,1,2}],
    5];

Now your 3 counterexamples behave thus:

Cross[{-u1,u2,u3},{v1,v2,v3}]/.crossrule2

myCross[{u1,-u2,-u3},{-v1,-v2,-v3}]

Cross[{u1,-u2,u3},{v1,v2,v3}]/.crossrule2

myCross[{-u1,u2,-u3},{-v1,-v2,-v3}]

Cross[{u1,u2,-u3},{v1,v2,v3}]/.crossrule2

myCross[{-u1,-u2,u3},{-v1,-v2,-v3}]

I agree with your final two remarks about the non-uniqueness of the 
representation. However, I think the original question was to do with 
spotting a particular type of pattern in the algebra, and then rewriting it 
in a more compact and familiar notation. Because this sort of 
pseudo-inversion is non-unique I tend to make up the replacement rules on an 
ad hoc basis as I said earlier.

Steve Luttrell

"DrBob" <drbob at bigfoot.com> wrote in message 
news:clndek$o4s$1 at smc.vnet.net...
> This won't always work, however.
>
> crossrule = {a3_ b2_ - a2_ b3_, a1_ b3_ - a3_ b1_,
>   a2_ b1_ - a1_ b2_} -> myCross[{b1, b2, b3}, {a1, a2, a3}];
>
> Cross[{u1, u2, u3}, {v1, v2, v3}] /. crossrule
> myCross[{u1,u2,u3},{v1,v2,v3}]
>
> Fine so far, but these should return cross products too:
>
> Cross[{-u1,u2,u3},{v1,v2,v3}]/.crossrule
> {-u3 v2+u2 v3,u3 v1+u1 v3,-u2 v1-u1 v2}
>
> Cross[{u1,-u2,u3},{v1,v2,v3}]/.crossrule
> {-u3 v2-u2 v3,u3 v1-u1 v3,u2 v1+u1 v2}
>
> Cross[{u1,u2,-u3},{v1,v2,v3}]/.crossrule
> {u3 v2+u2 v3,-u3 v1-u1 v3,-u2 v1+u1 v2}
>
> Besides this problem there are other issues, such as:
>
> (1) EVERY vector is the cross product of two vectors,
>
> and
>
> (2) The vectors are never unique.
>
> Good luck!!
>
> Bobby
>
> On Sat, 23 Oct 2004 00:22:49 -0400 (EDT), Steve Luttrell 
> <steve_usenet at _removemefirst_luttrell.org.uk> wrote:
>
>> You can use a replacement rule to convert your expression into the 
>> compact
>> form that you want:
>>
>> Define the replacement rule.
>>
>> crossrule = {a3_ b2_ -a2_ b3_,a1_ b3_ -a3_ b1_,a2_ b1_ -a1_ b2_} ->
>> myCross[{b1,b2,b3},{a1,a2,a3}];
>>
>> Apply the rule to your expression.
>>
>> result = {-u3 v2 + u2 v3, u3 v1 - u1 v3, -u2 v1 + u1 v2}/.crossrule
>>
>> which gives the output
>>
>> myCross[{u1,u2,u3},{v1,v2,v3}]
>>
>> You could make the notation more compact by doing this
>>
>> result /. {{u1,u2,u3} -> u,{v1,v2,v3} -> v}
>>
>> which gives the output
>>
>> myCross[u,v]
>>
>> Another (related but more sophisticated) method is to use the
>> Utilities`Notation` package to implement the reformatting of your output
>> automatically. This includes a way of defining infix operators so that 
>> you
>> can get an output that looks like u x v rather than myCross[u,v]
>>
>> Steve Luttrell
>>
>> "melon" <sweetmelon at gmail.com> wrote in message
>> news:cl9t1d$7m0$1 at smc.vnet.net...
>>> I used mathmetica5 to solve a vector equation. Mathmetica expands the
>>> result into sub-variable form. But I want to get the vector form. How
>>> to do that?
>>>
>>> ex:
>>> Solution:
>>> {-u3 v2 + u2 v3, u3 v1 - u1 v3, -u2 v1 + u1 v2}
>>> I want mathematica give me {u cross v}
>>>
>>> p,s,  Could you add me into the forum? Thank you very much. Regards.
>>>
>>> ShenLei
>>>
>>
>>
>>
>>
>>
>
>
>
> -- 
> DrBob at bigfoot.com
> www.eclecticdreams.net
> 



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