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From: William Elliot <ma...@panix.com>
Newsgroups: sci.math
Subject: Re: Basic question on line and plane
Date: Wed, 22 Feb 2012 02:42:07 -0800
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On Wed, 22 Feb 2012, Nayna wrote:
> On Feb 22, 2:52 pm, William Elliot <ma...@panix.com> wrote:
> > On Wed, 22 Feb 2012, Nayna wrote:

> > > If I have two points in 3 dimensions lying in different planes eg.
> > > (1,1,1) and (-3, 8, -2).
> >
> > Every two points lie in infinitely many planes and in addition
> > those two points can lie in infinitely many different pair of
> > planes.  Thus the condition of lying in different planes is
> > irrevalent.
> >
> > > Will these two points form a line or a plane ?
> >
> > They will determine one unique line and will select
> > out of all possible planes, the infinite collection
> > of planes that contain the line determined by the two
> > points.
> >
> > > General concepts says that in 3 dimensional it is always plane and 2
> > > dimensional it is always line.
> >
> > Huh?
> >
> > > Secondly, only non-collinear points form a plane. collinear points
> > > always form a line.
> >
> > Thre non-collinear points determine a plane and any multi-point
> > set of collinear points by definition, all lie on same line.
> > That the line is unique is a simple theorem.
> >
> > > I have this confusion because :
> > > 1. These two points are in 3 dimensional.
> >
> > Are in 3 dimensional space.
> >
> > > 2. There are only two points so they can always be collinear.
> >
> > Yes.
> 
> Thanks for the clarity.
> So, is my this understanding wrong :
> 
> > > General concepts says that in 3 dimensional it is always plane and 2 
> > > dimensional it is always line.

That doesn't make sense.

> Aren't the equation ax + by + cz = d  always forms a plane , if it
> gets satisfied and a = b = c = d != 0.
> 
a = b = c = d /= 0 means
a = b and b = c and c = d and d /= 0.

That's correct but not all planes.  It excludes, for example, 
planes through the orgin and the coordinate planes.


{ (x,y,z) | ax + by + cz = d } is a plane iff