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MathGroup Archive 2002

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Re: Re: Generating Two Unit Orthogonal Vectors (correction)

  • To: mathgroup at smc.vnet.net
  • Subject: [mg36486] Re: [mg36428] Re: Generating Two Unit Orthogonal Vectors (correction)
  • From: Gianluca Gorni <gorni at dimi.uniud.it>
  • Date: Mon, 9 Sep 2002 00:29:23 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

The vectors that NullSpace gives do not need to be orthogonal to
each other. What about the following variant, that adds one vector
at a time? It may not be fast enough for serious number crunching.

OrthogonalComplement[v__ /; MatrixQ[{v}]] :=
  With[{n = Length[NullSpace[{v}]]},
    Take[Nest[Join[{First[NullSpace[#]]}, #] &, {v}, n], n]];

OrthonormalComplement[v__ /; MatrixQ[{v}]] :=
  Map[#/Sqrt[#.#] &, OrthogonalComplement[v]];

When the input of OrthogonalComplement is integer, I expect the output
to be integer too:

OrthogonalComplement[{1, 2, 3}]

{{1, -5, 3}, {-3, 0, 1}}

                        Gianluca Gorni


> From: "Ersek, Ted R" <ErsekTR at navair.navy.mil>
To: mathgroup at smc.vnet.net
> Date: Fri, 6 Sep 2002 03:17:01 -0400 (EDT)
> Subject: [mg36486] [mg36428] Re: Generating Two Unit Orthogonal Vectors (correction)
> 
> OrthogonalUnitVectors  that I sent a few minutes ago doesn't do what I
> thought it would.
> After making either definition below I get the following:
> 
> In[2]:=
> s1=OrthogonalUnitVectors[{1,0,1/2,1,0},{0,1,-1,1/2,2}]
> 
> Out[2]=
> {{0, -2/Sqrt[5], 0, 0, 1/Sqrt[5]}, {-2/3, -1/3, 0, 2/3, 0}, {-1/3, 2/3,
> 2/3, 0, 0}}
> 
> 
> The dot products below aren't zero, so the vectors aren't orthogonal.  What
> went wrong?
> 
> 
> In[3]:= 
> Part[s1,1].Part[s1,2]
> 
> Out[3]=
> 2/(3*Sqrt[5])
> 
> 
> In[4]:= 
> Part[s1,1].Part[s1,3]
> 
> Out[3]=
> -4/(3*Sqrt[5])
> 
> 
> --------------
>> Hugh Goyder and David Park gave a most elegant function to find two
>> vectors that are orthogonal to one vector in 3D.  The key to coming up
>> with the elegant solution is an understanding of Mathematica's NullSpace
>> function.  We can easily make the version from Hugh and David much more
>> general with the version below.
>> -------------
>> OrthogonalUnitVectors[vect__?VectorQ]:=
>> #/Sqrt[#.#]&/@NullSpace[{vect}]
>> 
>> -------------
>> The version above will give a set of unit orthogonal vectors if given any
>> number of vectors in any dimension.
>> So besides giving it a 3D vector we can give it the following:
>> OrthogonalUnitVectors[{2,1,0,-1,1}]
>> or
>> OrthogonalUnitVectors[{0,1,0,1/2,1},{1,0,-1,1/2}]
>> 
>> ------------
>> But the short version above isn't very robust.
>> (1)  Clear[x,y,z];NullSpace[{{x,y,z}}]
>> returns two vectors orthogonal to {x,y,z}, but the two vectors
>> NullSpace returns aren't orthogonal to each other.
>> So (OrthogonalUnitVectors) should only work with numeric vectors.
>> 
>> (2)  We should ensure all the vectors have the same dimension and length
>>> 1.
>> 
>> I give a less concise version below that corrects these problems.
>> ------------
>> 
>> OrthogonalUnitVectors[vect__?(VectorQ[#,NumericQ]&)]/;
>> (SameQ@@Length/@{vect})&&(Length[First[{vect}]]>1):=
>> #/Sqrt[#.#]&/@NullSpace[{vect}]
>> 
>> --------------
>> Regards,
>> Ted Ersek
>> Get Mathematica tips, tricks from
>> http://www.verbeia.com/mathematica/tips/Tricks.html
>> 
> 



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