Re: Q. about Solve as applied to vector equations
- To: mathgroup@smc.vnet.net
- Subject: [mg12177] Re: Q. about Solve as applied to vector equations
- From: Paul Abbott <paul@physics.uwa.edu.au>
- Date: Fri, 1 May 1998 03:08:29 -0400
- Organization: University of Western Australia
- References: <6hpb4f$d43@smc.vnet.net>
Rajarishi S Sinha wrote:
> I'm trying to solve the following vector expression for the vector 'r':
>
> x*n1 X r1 + (1-x)*n2 X r2 = (1/norm2(x*n1 + (1-x)*n2))*(x+n1 + (1-x)*n2)
> X r 0<=x<=1
I think there an error in this expression. Should it not read
x*n1 X r1 + (1-x)*n2 X r2 = (1/norm2(x*n1 + (1-x)*n2))*(x*n1 +
(1-x)*n2) X r
Since you can work with typeset expressions in Mathematica Notebooks
perhaps the following is a better way of expressing this:
Cell[BoxData[\(TraditionalForm
\`x\ n\_1\[Cross]r\_1 + \ \((1 - x)\) n\_2\[Cross]\ r\_2 ==
\(\(x\ n\_1 + \((1 - x)\)\ n\_2\
\)\/\[LeftDoubleBracketingBar]x\ n\_1 +
\((1 - x)\) n\_2\[RightDoubleBracketingBar]\)\[Cross]\
r\)],
"Input"]
(select Cell[...] and paste into a Mathematica Notebook to view this
Cell).
As far as I can see, what you are trying to do is, for fixed x, n1, n2,
r1, r2, effectively solve the vector equation a = x X r for r with a
and x supplied. In terms of components,
In[1]:= eqn = {a, b, c} == Cross[{x, y, z}, {r, s, t}] Out[1]=
{a,b,c}=={t y-s z,r z-t x,s x-r y}
Solve does not work
In[2]:= Solve[%,{r,s,t}]
Out[2]= {}
for a good reason. If you eliminate r and s,
In[3]:= Eliminate[%%,{r,s}]
Out[3]= c z==-a x-b y
you see that there, for a solution to exist, there needs to be a
relationship between the vectors a={a,b,c} and x={x,y,z}. If this
relationship holds, then
In[4]:= eqn/.First[%]
Out[4]=
s (a x + b y) r (a x + b y)
{a, b, c} == {t y + -------------, -t x - -------------, s x - r y}
c c
and you can now solve for two of the components of r
In[5]:= Solve[%,{r,s}]
Out[5]=
b c + t x c c (a - t y)
{{r -> -(-----------), s -> -----------}}
a x + b y a x + b y
Cheers,
Paul
____________________________________________________________________
Paul Abbott Phone: +61-8-9380-2734
Department of Physics Fax: +61-8-9380-1014
The University of Western Australia Nedlands WA 6907
mailto:paul@physics.uwa.edu.au AUSTRALIA
http://www.pd.uwa.edu.au/~paul
God IS a weakly left-handed dice player
____________________________________________________________________