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Re: Are points co-planar in (numDimensions-1)?

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  • Subject: [mg43240] Re: Are points co-planar in (numDimensions-1)?
  • From: "AngleWyrm" <no_spam_anglewyrm at hotmail.com>
  • Date: Wed, 20 Aug 2003 22:26:27 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

My last post on this subject lacked depth, so here's more info.

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 have a dataset I wish to test for this property in various dimensions:

numDimensions = 2;
dataSet = ReadList["data.txt", Number ];
dataSet = Partition[ dataSet, numDimensions ];

At this point I have a set of 2D vectors, and if I take any two of them they HAVE to be coplanar,
right?

sample = Take[ dataSet, numDimensions];
Sum[ k\_i sample[[i]], {i, numDimensions} ]    (I've used \_ to indicate subscript here)

Now my problem is in solving for zero on the last equation. I've tried like so:
Solve[ Sum[ k\_i sample[[i]], {i, numDimensions} ] ==0, {k\_i} ]

but it gives an empty set every time. Thanks for any help.
-Jonathan


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