Re: Are points co-planar in (numDimensions-1)?
- To: mathgroup at smc.vnet.net
- Subject: [mg43327] Re: Are points co-planar in (numDimensions-1)?
- From: "AngleWyrm" <no_spam_anglewyrm at hotmail.com>
- Date: Sun, 24 Aug 2003 04:55:41 -0400 (EDT)
- References: <bi195e$akp$1@smc.vnet.net> <bi7nu3$pc8$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
"Olaf Rogalsky" <Olaf.Rogalsky at physik.uni-erlangen.de> wrote in message news:bi7nu3$pc8$1 at smc.vnet.net... > AngleWyrm wrote: > > Given some n-dimensional vectors, are they coplanar in n-1? Let a1, a2, ..., an be vectors. If they > > are coplanar, then there exists a set of coefficients {k1, k2, ..., kn}, not all zero, which satisfy > > the equation: k1 a1 + k2 a2 + ... + kn an = 0. > I don't agree. This is the definition of linear independence of n vectors, not of coplanarity. > A set of k vectors {a_i in Vn | i=1..k} in a n-dimensional vector space Vn are said to be coplanar, > if all k vectors {a_i in Vn | i=1..k} are elements of a (n-1)-dimensional affine subspace U of Vn. If a and b are non-parallel vectors (and not 0), and c is coplanar with a and b, then it is possible to transform to c from a and b like so: c = k1a + k2b // where k1 and k2 are some factors. k1a + k2b - c = 0 // rearranging the equation by subracting c Otherwise, if a and b are parallel, there is some factor so that say, b=k3a. Then k3a - b = 0 // also by subtraction Thus, to summarize, if a,b, and c are coplanar (and not 0), then there exists some relation of the form: k1 a + k2 b + k3 c = 0 > Solve[ Sum[ k\_i sample[[i]], {i, numDimensions} ] ==0, Table[k\_i,{i,1,numDimensions}] ] Thank you, this helped.