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MathGroup Archive 2002

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Re: Generating Two Unit Orthogonal Vectors

  • To: mathgroup at smc.vnet.net
  • Subject: [mg36426] Re: Generating Two Unit Orthogonal Vectors
  • From: "Ersek, Ted R" <ErsekTR at navair.navy.mil>
  • Date: Fri, 6 Sep 2002 03:16:58 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

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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