[R] How to get the normal direction to a plane?
Fred
fzh113 at hecky.it.northwestern.edu
Sat Jul 3 00:35:02 CEST 2004
Thanks a lot, Spencer
Fred
----- Original Message -----
From: "Spencer Graves" <spencer.graves at pdf.com>
To: "Fred" <fzh113 at hecky.it.northwestern.edu>
Cc: "'R-help'" <R-help at stat.math.ethz.ch>
Sent: Friday, July 02, 2004 5:18 PM
Subject: Re: [R] How to get the normal direction to a plane?
> While we need 3 points to determine a line, we need only 2
> vectors, provided they both have the same origin and differ in direction
> not just magnitude; this latter condition is the same as saying that
> the 3 points can not lie on a line.
>
> To apply this, suppose a, b, and c are 3 vectors in k-space, and
> let X = the k x 2 matrix with columns b-a and c-a. By the assumption
> that the three points do not lie on a line, the matrix X has rank 2, so
> X'X is nonsingular. Let P = X*inv(X'X)X'. Note that P is idempotent,
> i.e., P*P = P. Further, note that Pz is a vector in the column space of
> X, for any k-vector z. Further, (I-P) is also idempotent and projects
> any vector onto the subspace orthogonal to P. Thus, (I-P)z will be
> orthogonal to P and therefore also orthogonal to X, for any k-vector z.
>
> This discussion reveals a subtle flaw in the logic as stated
> (which I didn't see until I worked the exercise): Only in the case
> where k = 3 is there only one direction that is orthogonal to this
> plane. In general, there are (k-2) such directions. For more
> information, see any good book on finite dimensional vector spaces such
> as Halmos (1974), or Google this or see ?svd or ?qr or the references
> cited therein.
>
> hope this helps. spencer graves
>
> Fred wrote:
>
> >Dear All
> >
> >Maybe the following is a stupid question.
> >Assume I have 3 coordinate points (not limited to be in 2D or 3D space)
> >a, b, c.
> >It is known that these 3 points will define a plane.
> >The problem is how to get the normal direction that is orthogonal to
> >this plane.
> >
> >Is there an easy way to calculate it using the values of a, b, and c?
> >
> >Thanks for any point or help on this.
> >
> >Fred
> >
> > [[alternative HTML version deleted]]
> >
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> >
>
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