Services & Resources / Wolfram Forums
-----
 /
MathGroup Archive
2002
*January
*February
*March
*April
*May
*June
*July
*August
*September
*October
*November
*December
*Archive Index
*Ask about this page
*Print this page
*Give us feedback
*Sign up for the Wolfram Insider

MathGroup Archive 2002

[Date Index] [Thread Index] [Author Index]

Search the Archive

Fwd: Generating Two Unit Orthogonal Vectors to a 3D Vector

  • To: mathgroup at smc.vnet.net
  • Subject: [mg36374] Fwd: [mg36352] Generating Two Unit Orthogonal Vectors to a 3D Vector
  • From: Garry Helzer <gah at math.umd.edu>
  • Date: Wed, 4 Sep 2002 02:56:28 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

>
> OrthogonalUnitVectors::usage = "OrthogonalUnitVectors[v:{_,_,_}] will 
> return
> two unit vectors orthogonal to each other and to v."
>
> You can assume that v is nonzero.
>
> David Park
> djmp at earthlink.net
> http://home.earthlink.net/~djmp/

Well the Mathematica routine cannot be more elegant than the 
mathematics. There are an infinite number of solutions and so an 
arbitrary choice must be made.

I will omit the unit condition because orthogonal sets of integral 
vectors are useful in examples and exercises --and I prefer to do 
graphics in a projective context where "orthogonal" is well defined but 
"unit" is not. For unit vectors just map the output with something like 
#/Sqrt[#.#]&.

The vector v=(a1,a2,a3) is nonzero if and only if at most one of the 
vectors (a2,-a1,0), (a3,0,-a1), or (0,a3,-a2) is zero.  These vectors 
are all orthogonal to v. But which one to choose? For numerical 
stability we should probably take the vector largest in some norm. The 
1-norm, Plus@@Abs[#]&, is easy to compute. Having picked the second 
vector find the third with a cross product. This assures a right-handed 
system.

OrthogonalVectors[v : {a1_, a2_, a3_}] :=
    With[{w =
         First[Sort[{{a2, -a1, 0}, {a3, 0, -a1}, {0, a3, -a2}},
             OrderedQ[{Plus @@ Abs[#2], Plus @@ Abs[#1]}] &]]},
     {w, Cross[v, w]}
     ]

Garry Helzer
Department of  Mathematics
University of Maryland
1303 Math Bldg
College Park, MD 20742-4015



  • Prev by Date: Re: Generating Two Unit Orthogonal Vectors to a 3D Vector
  • Next by Date: Re: Generating Two Unit Orthogonal Vectors to a 3D Vector
  • Previous by thread: Re:Generating Two Unit Orthogonal Vectors to a 3D Vector
  • Next by thread: Re: Generating Two Unit Orthogonal Vectors to a 3D Vector