From: Fred <fzh113_at_hecky.it.northwestern.edu>

Date: Sat 03 Jul 2004 - 08:35:02 EST

R-help@stat.math.ethz.ch mailing list

https://www.stat.math.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html Received on Sat Jul 03 08:39:53 2004

Date: Sat 03 Jul 2004 - 08:35:02 EST

Thanks a lot, Spencer

Fred

----- Original Message -----

From: "Spencer Graves" <spencer.graves@pdf.com>
To: "Fred" <fzh113@hecky.it.northwestern.edu>
Cc: "'R-help'" <R-help@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]]
**> >
**> >______________________________________________
**> >R-help@stat.math.ethz.ch mailing list
**> >https://www.stat.math.ethz.ch/mailman/listinfo/r-help
**> >PLEASE do read the posting guide!
*

http://www.R-project.org/posting-guide.html

*> >
**> >
**>
*

R-help@stat.math.ethz.ch mailing list

https://www.stat.math.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html Received on Sat Jul 03 08:39:53 2004

*
This archive was generated by hypermail 2.1.8
: Wed 03 Nov 2004 - 22:54:40 EST
*