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question help pls

How can i answer this question?
(edited 3 years ago)

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Avatar for mqb2766
mqb2766
21
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.
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 p1,p2\mathbf{p}_1,\mathbf{p}_2 be vectors representing the points P1,P2.

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

p1⋅n=d\mathbf{p}_1 \cdot \mathbf{n} = d
p2⋅n=d\mathbf{p}_2 \cdot \mathbf{n} = d

  ⟹  (p2−p1)⋅n=0\implies (\mathbf{p}_2 - \mathbf{p}_1) \cdot \mathbf{n} = 0

The line through P1,P2 is given as r=p1+t(p2−p1)\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.
(edited 3 years ago)
Original post by ottersandseals1
Something like this?
Attachment not found
Attachment not found

@RDKGames is this correct?
Reply 4
Avatar for mqb2766
mqb2766
21
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.
Original post by RDKGames
Let p1,p2\mathbf{p}_1,\mathbf{p}_2 be vectors representing the points P1,P2.

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

p1⋅n=d\mathbf{p}_1 \cdot \mathbf{n} = d
p2⋅n=d\mathbf{p}_2 \cdot \mathbf{n} = d

  ⟹  (p2−p1)⋅n=0\implies (\mathbf{p}_2 - \mathbf{p}_1) \cdot \mathbf{n} = 0

The line through P1,P2 is given as r=p1+t(p2−p1)\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?
Reply 6
Avatar for mqb2766
mqb2766
21
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.
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
Reply 8
Avatar for mqb2766
mqb2766
21
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.
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)?
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.
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
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:
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?
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?
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?
Original post by ottersandseals1
Sorry which threads are you talking about?

https://www.thestudentroom.co.uk/showthread.php?t=6755400
https://www.thestudentroom.co.uk/showthread.php?t=6733548
https://www.thestudentroom.co.uk/showthread.php?t=6732568
https://www.thestudentroom.co.uk/showthread.php?t=6732154
https://www.thestudentroom.co.uk/showthread.php?t=6686314

etc ...
Original post by ottersandseals1
Something like this?
Attachment not found
Attachment not found


Your handwriting is quite messy. I would sort this out before working with planes and trying to become a pilot.
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?
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?
(edited 3 years ago)

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