ottersandseals1
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How can i answer this question?
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mqb2766
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(Original post by ottersandseals1)
How can i answer this question? Been stuck on for a while.

If a plane contains two distinct points P1 and P2, show that it contains every point on the line through P1 and P2.

Any help is appreciated
Use the definition of a p!ane and write down the conditions for P1 and P2.
Interpret the line segment as a linear combination of P1 and P2 and Interpret the segment in terms of the two points.
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RDKGames
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(Original post by ottersandseals1)
How can i answer this question? Been stuck on for a while.

If a plane contains two distinct points P1 and P2, show that it contains every point on the line through P1 and P2.

Any help is appreciated
Let \mathbf{p}_1,\mathbf{p}_2 be vectors representing the points P1,P2.

A plane with general equation \mathbf{r}\cdot \mathbf{n} = d contains these two points, therefore we have that

\mathbf{p}_1 \cdot \mathbf{n} = d
\mathbf{p}_2 \cdot \mathbf{n} = d

\implies (\mathbf{p}_2 - \mathbf{p}_1) \cdot \mathbf{n} = 0

The line through P1,P2 is given as \mathbf{r} = \mathbf{p}_1 + t (\mathbf{p}_2 - \mathbf{p}_1)

Show that this satisfies the plane's equation, which would hence imply that this line lies entirely in the plane.
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ottersandseals1
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(Original post by ottersandseals1)
Something like this?
Attachment 970884 Attachment 970886
RDKGames is this correct?
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mqb2766
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(Original post by ottersandseals1)
RDKGames is this correct?
You shouldn't assume the p!ane/first point goes through the origin .
The previous post is a near full solution.
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ottersandseals1
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(Original post by RDKGames)
Let \mathbf{p}_1,\mathbf{p}_2 be vectors representing the points P1,P2.

A plane with general equation \mathbf{r}\cdot \mathbf{n} = d contains these two points, therefore we have that

\mathbf{p}_1 \cdot \mathbf{n} = d
\mathbf{p}_2 \cdot \mathbf{n} = d

\implies (\mathbf{p}_2 - \mathbf{p}_1) \cdot \mathbf{n} = 0

The line through P1,P2 is given as \mathbf{r} = \mathbf{p}_1 + t (\mathbf{p}_2 - \mathbf{p}_1)

Show that this satisfies the plane's equation, which would hence imply that this line lies entirely in the plane.
Sorry, where does the t come from?
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mqb2766
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(Original post by ottersandseals1)
Sorry, where does the t come from?
t is a free variable between 0 and 1. It generates all points on the line between P1 and P2.
If you can show all points (for all t) lie on the plane, then you're done.
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ottersandseals1
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(Original post by mqb2766)
t is a free variable between 0 and 1. It generates all points on the line between P1 and P2.
If you can show all points (for all t) lie on the plane, then you're done.
Do i use the equation r.n = d ? Sorry if it's taking me a while to get
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mqb2766
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(Original post by ottersandseals1)
Do i use the equation r.n = d ? Sorry if it's taking me a while to get
That is an equation of the plane. So yes.
r is the point, n is the (unit) normal and d is the distance from the origin.
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ottersandseals1
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(Original post by mqb2766)
That is an equation of the plane. So yes.
r is the point, n is the (unit) normal and d is the distance from the origin.
And from the example above R = (P2 - P1)?
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RDKGames
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(Original post by mqb2766)
t is a free variable between 0 and 1. It generates all points on the line between P1 and P2.
If you can show all points (for all t) lie on the plane, then you're done.
t does not necessarily need to be between 0 and 1.

(Original post by mqb2766)
That is an equation of the plane. So yes.
r is the point, n is the (unit) normal and d is the distance from the origin.
n is not necessarily the unit normal.
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mqb2766
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(Original post by ottersandseals1)
And from the example above R = (P2 - P1)?
r is either p1 or p2. They satisfy r.n=d
But subtracting the two equations shows the difference
(P1-P2).n = 0
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RDKGames
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OP, out of curiosity, why did you remove all your posts on your previous threads? Looks a bit dodgy .. almost as if you asked for help with assessed work and want to get rid of the evidence :holmes:
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ottersandseals1
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(Original post by RDKGames)
t does not necessarily need to be between 0 and 1.


n is not necessarily the unit normal.
(Original post by mqb2766)
r is either p1 or p2. They satisfy r.n=d
But subtracting the two equations shows the difference
(P1-P2).n = 0
Does that mean t can be n then?
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ottersandseals1
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(Original post by RDKGames)
OP, out of curiosity, why did you remove all your posts on your previous threads? Looks a bit dodgy .. almost as if you asked for help with assessed work and want to get rid of the evidence :holmes:
Sorry which threads are you talking about?
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mqb2766
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(Original post by ottersandseals1)
Does that mean t can be n then?
t is a number [0,1], n is the (unit) normal vector.
Which equation of a plane are you happy using?
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Sexy Jeddah
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(Original post by ottersandseals1)
Something like this?
Attachment 970884 Attachment 970886
Your handwriting is quite messy. I would sort this out before working with planes and trying to become a pilot.
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RDKGames
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(Original post by ottersandseals1)
Sorry which threads are you talking about?
https://www.thestudentroom.co.uk/sho....php?t=6755400
https://www.thestudentroom.co.uk/sho....php?t=6733548
https://www.thestudentroom.co.uk/sho....php?t=6732568
https://www.thestudentroom.co.uk/sho....php?t=6732154
https://www.thestudentroom.co.uk/sho....php?t=6686314

etc ...
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mqb2766
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(Original post by ottersandseals1)
I think r.n = d is the best, unless you think i could use a better one?
Sure, but you don't seem to understand it?
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mqb2766
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(Original post by ottersandseals1)
So, a is a known vector for a fixed point in a plane, n is the normal vector and r is a resultant vector?
r.n = d
r is a point (vector) on the plane. Not sure what a or the resultant vector are?
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