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Basic question on line and plane
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Nayna
Feb 22, 2012, 4:02:41 AM2/22/12
to
Hi all,
I have one basic question to understand the different between line and
plane.
If I have two points in 3 dimensions lying in different planes eg.
(1,1,1) and (-3, 8, -2).
Will these two points form a line or a plane ?
General concepts says that in 3 dimensional it is always plane and 2
dimensional it is always line.
Secondly, only non-collinear points form a plane. collinear points
always form a line.
I have this confusion because :
1. These two points are in 3 dimensional.
2. There are only two points so they can always be collinear.
Please correct me where I went wrong and explain the right concept of
this difference ?
Thanks & Regards,
- Nayna
I have one basic question to understand the different between line and
plane.
If I have two points in 3 dimensions lying in different planes eg.
(1,1,1) and (-3, 8, -2).
Will these two points form a line or a plane ?
General concepts says that in 3 dimensional it is always plane and 2
dimensional it is always line.
Secondly, only non-collinear points form a plane. collinear points
always form a line.
I have this confusion because :
1. These two points are in 3 dimensional.
2. There are only two points so they can always be collinear.
Please correct me where I went wrong and explain the right concept of
this difference ?
Thanks & Regards,
- Nayna
Tonico
Feb 22, 2012, 4:38:27 AM2/22/12
to
On Feb 22, 11:02 am, Nayna <jainna...@gmail.com> wrote:
> Hi all,
>
> I have one basic question to understand the different between line and
> plane.
>
> If I have two points in 3 dimensions lying in different planes eg.
> (1,1,1) and (-3, 8, -2).
>
> Will these two points form a line or a plane ?
**** Your question is ill-posed: two general points in the real 3D
> Hi all,
>
> I have one basic question to understand the different between line and
> plane.
>
> If I have two points in 3 dimensions lying in different planes eg.
> (1,1,1) and (-3, 8, -2).
>
> Will these two points form a line or a plane ?
space (I'm assuming this is what you meant) "are not in different
planes" NATURALLY. They are inf YOU define two planes containing each
one point and you show these planes are different.
Second, two points in real 3D space "never form" neither a line nor a
plane: there is always one UNIQUE line that passes through both
planes, and there is always a plane (and, in general, infinite many
planes) that pass through both points.
In these geometric matters it is important to be pitch clear about
what one means.
Tonio
William Elliot
Feb 22, 2012, 4:52:50 AM2/22/12
to
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
> If I have two points in 3 dimensions lying in different planes eg.
> (1,1,1) and (-3, 8, -2).
>
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 ?
>
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.
> Secondly, only non-collinear points form a plane. collinear points
> always form a line.
>
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.
> 2. There are only two points so they can always be collinear.
>
Nayna
Feb 22, 2012, 5:02:42 AM2/22/12
to
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.
>
gets satisfied and a = b = c = d != 0.
William Elliot
Feb 22, 2012, 5:42:07 AM2/22/12
to
> 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.
> So, is my this understanding wrong :
>
> > > General concepts says that in 3 dimensional it is always plane and 2
> > > dimensional it is always line.
> Aren't the equation ax + by + cz = d always forms a plane , if it
> gets satisfied and a = b = c = d != 0.
>
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
a^2 + b^2 + c^2 /= 0.
Nayna
Feb 22, 2012, 9:14:32 AM2/22/12
to
What should be the approach to see that whether the combination of
vectors form line or plane.
I understand drawing row or column picture but they also becomes
difficult after 2 unknowns.
What all conditions should we take care in doing this proof.
For eg.
* Compute u + v + w and 2u + 2v + w. How do you know that u, v, w lie
in a plane ?
How should I start for finding the solution for this and in general
any equation ?
Thanks & Regards,
- Nayna Jain
Frederick Williams
Feb 22, 2012, 11:00:51 AM2/22/12
to
Nayna wrote:
> What all conditions should we take care in doing this proof.
> For eg.
> * Compute u + v + w and 2u + 2v + w. How do you know that u, v, w lie
> in a plane ?
> How should I start for finding the solution for this and in general
> any equation ?
If one of u, v, w is a linear combination of the other two, then they
> What all conditions should we take care in doing this proof.
> For eg.
> * Compute u + v + w and 2u + 2v + w. How do you know that u, v, w lie
> in a plane ?
> How should I start for finding the solution for this and in general
> any equation ?
lie in a plane.
I see no equation.
--
When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting
Ken Pledger
Feb 22, 2012, 5:07:19 PM2/22/12
to
In article
<e8ee850e-b97b-4b26...@j8g2000yqm.googlegroups.com>,
Nayna <jain...@gmail.com> wrote:
> ....
There has been some confusion because you have stated only part of
the problem. For example, what you have written above doesn't say what
u, v and w are. People can help you much better if you type out the
exact wording of the whole question which you're trying to answer.
Ken Pledger.
<e8ee850e-b97b-4b26...@j8g2000yqm.googlegroups.com>,
Nayna <jain...@gmail.com> wrote:
> ....
> What all conditions should we take care in doing this proof.
> For eg.
> * Compute u + v + w and 2u + 2v + w. How do you know that u, v, w lie
> in a plane ?
> How should I start for finding the solution for this and in general
> any equation ? ....
> For eg.
> * Compute u + v + w and 2u + 2v + w. How do you know that u, v, w lie
> in a plane ?
> How should I start for finding the solution for this and in general
There has been some confusion because you have stated only part of
the problem. For example, what you have written above doesn't say what
u, v and w are. People can help you much better if you type out the
exact wording of the whole question which you're trying to answer.
Ken Pledger.
Nayna
Feb 23, 2012, 12:21:54 AM2/23/12
to
On Feb 23, 3:07 am, Ken Pledger <ken.pled...@vuw.ac.nz> wrote:
> In article
> <e8ee850e-b97b-4b26-97c8-5b207f18f...@j8g2000yqm.googlegroups.com>,
Here is the complete question :
Compute u + v + w and 2u + 2v +w. How do you know u, v, w lie in a
plane ?
In a plane u = 1 v = -3 w = 2
2 1 -3
3 -2 -1
Couldn't put brackets here for the column vectors in plain text.
> In article
> <e8ee850e-b97b-4b26-97c8-5b207f18f...@j8g2000yqm.googlegroups.com>,
>
> Nayna <jainna...@gmail.com> wrote:
> > ....
> > What all conditions should we take care in doing this proof.
> > For eg.
> > * Compute u + v + w and 2u + 2v + w. How do you know that u, v, w lie
> > in a plane ?
> > How should I start for finding the solution for this and in general
> > any equation ? ....
>
> There has been some confusion because you have stated only part of
> the problem. For example, what you have written above doesn't say what
> u, v and w are. People can help you much better if you type out the
> exact wording of the whole question which you're trying to answer.
>
> Ken Pledger.
Ok. Yes. Sorry for that !!
> Nayna <jainna...@gmail.com> wrote:
> > ....
> > What all conditions should we take care in doing this proof.
> > For eg.
> > * Compute u + v + w and 2u + 2v + w. How do you know that u, v, w lie
> > in a plane ?
> > How should I start for finding the solution for this and in general
> > any equation ? ....
>
> There has been some confusion because you have stated only part of
> the problem. For example, what you have written above doesn't say what
> u, v and w are. People can help you much better if you type out the
> exact wording of the whole question which you're trying to answer.
>
> Ken Pledger.
Here is the complete question :
Compute u + v + w and 2u + 2v +w. How do you know u, v, w lie in a
plane ?
In a plane u = 1 v = -3 w = 2
2 1 -3
3 -2 -1
Couldn't put brackets here for the column vectors in plain text.
Tonico
Feb 23, 2012, 1:26:29 AM2/23/12
to
Ah, why didn't you put the question in its original form from the
beginning? Now, two non linearly dependent vectors (this means: two
vectors none of which is a scalar multiple of the other) always
determine a plane, and a third one belongs to that plane iff it is
linearly DEPENDENT on those two.
For example, we can easily see that w = -u - v ...
Tonio
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