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MathGroup Archive 2002

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RE: RE: Generating Two Unit Orthogonal Vectors

  • To: mathgroup at
  • Subject: [mg36460] RE: [mg36429] RE: Generating Two Unit Orthogonal Vectors
  • From: "DrBob" <drbob at>
  • Date: Sat, 7 Sep 2002 02:54:52 -0400 (EDT)
  • Reply-to: <drbob at>
  • Sender: owner-wri-mathgroup at

I checked again the six solutions I had previously timed, and they DO
give orthogonal results.  (None of them depend on NullSpace for that.)

By the way, I reused my "combinations" function (from a recent problem
on adding fractions to get 1) to check for orthogonality:

    orthogonalQ[v : {__?VectorQ}] := And @@ (Chop@(
Dot @@ #) == 0 & /@ combinations[v, {2}])

    << DiscreteMath`Combinatorica`;
    r = Range[1, 9];
    combinations::usage = "combinations[list,n:{__Integer}] lists 
    the combinations of list taken n at a time";
    combinations[r_List, n_Integer, {}] := 
      If[n > Length@r, {}, DiscreteMath`Combinatorica`KSubsets[r, n]];
    combinations[r_List, n_Integer, e_?VectorQ] := Join[e, #] & /@ \
DiscreteMath`Combinatorica`KSubsets[Complement[r, e], n];
    combinations[r_List, n_Integer, e : {__?VectorQ}] := Flatten[
      combinations[r, n, #] & /@ e, 1];
    combinations[r_List, n : {__Integer}] := Which[Plus @@ n ==
Length@r, \
Join[#, Complement[r, #]] & /@ 
    combinations[r, Drop[n, -1]], Plus @@ n > Length@r, {}, True, Fold[
        combinations[r, #2, #1] &, {}, n]]


-----Original Message-----
From: David Park [mailto:djmp at] 
To: mathgroup at
Subject: [mg36460] [mg36429] RE: Generating Two Unit Orthogonal Vectors 

Daniel Lichtblau has pointed out that NullSpace does not generally give
orthogonal vectors. Therefore the routines that depended upon that were
error. He says that it does give orthogonal vectors when the input
contains approximate numbers. For graphical purposes this will be good
enough for me. Therefore I modify Ted's routine to

OrthogonalUnitVectors[vect__?(VectorQ[#, NumericQ] &)] /;
        (SameQ @@ Length /@ {vect}) && (Length[First[{vect}]] > 1) :=
    #/Sqrt[#.#] & /@ NullSpace[{vect}// N]

and the short version for 3D vectors

OrthogonalUnitVectors[v : {_, _, _}] := #/Sqrt[#.#] & /@

For exact vectors I might use for 3D

OrthogonalUnitVectors[v : {_, _, _}] :=
    #/Sqrt[#.#] & /@ {temp = First[NullSpace[{v}]], v\[Cross]temp}

I'm still looking for something that is easy to remember.

David Park
djmp at

From: Ersek, Ted R [mailto:ErsekTR at]
To: mathgroup at

Hugh Goyder and David Park gave a most elegant function to find two
that are orthogonal to one vector in 3D.  The key to coming up with the
elegant solution is an understanding of Mathematica's NullSpace
We can easily make the version from Hugh and David much more general
the version below.

The version above will give a set of unit orthogonal vectors if given
number of vectors in any dimension.
So besides giving it a 3D vector we can give it the following:

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

I give a less concise version below that corrects these problems.


   Ted Ersek
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