Finding where a tangent plane and a surface meet? Watch

_Caz_
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Hi I've just seen this come up on the Jun14 paper (fp3 ocr mei) and I can't find anything on the internet on how to do it. My teacher went through it in class and I still don't get it. The actual question was to find where the the plane

10x-y+2z=6

meets the surface

g(x,y,z)=x^2 + 3y^2 + 2z^2 + 2yz + 6xz - 4xy - 24 =0.

I understand that grad g has got to equal some multiple of the vector 10i-j+2k but don't know where to go from there.

Any help would be great thank you.

I cant find the mark scheme on the internet either or find any other questions like it.
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ghostwalker
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(Original post by _Caz_)
Any help would be great thank you.
So, you have:

grad g = k(10,-1,2)...............(1)

I'd use the equation of the plane to eliminate one of the variables in grad g.

Then equating coefficients in (1) you can eliminate k to get a couple of linear equations in two variables which are solveable as standard simultaneous equations.
(Let me know if you need me to expand on that)

Have to say, I must keep making slips as I can't get it to come out nicely.
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_Caz_
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(Original post by ghostwalker)
So, you have:

grad g = k(10,-1,2)...............(1)

I'd use the equation of the plane to eliminate one of the variables in grad g.

Then equating coefficients in (1) you can eliminate k to get a couple of linear equations in two variables which are solveable as standard simultaneous equations.
(Let me know if you need me to expand on that)

Have to say, I must keep making slips as I can't get it to come out nicely.
Thanks I'm going to try it again tomorrow. Sick of doing it now haha!

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atsruser
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(Original post by ghostwalker)
So, you have:

grad g = k(10,-1,2)...............(1)

I'd use the equation of the plane to eliminate one of the variables in grad g.

Then equating coefficients in (1) you can eliminate k to get a couple of linear equations in two variables which are solveable as standard simultaneous equations.
(Let me know if you need me to expand on that)

Have to say, I must keep making slips as I can't get it to come out nicely.
I had a go at this earlier, and the numbers seem to get unpleasantly large, at which point I gave up. I'm wondering if the question has been quoted incorrectly. It seems too tricky for a standard A level exam question as it stands.
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ghostwalker
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(Original post by atsruser)
I had a go at this earlier, and the numbers seem to get unpleasantly large, at which point I gave up. I'm wondering if the question has been quoted incorrectly. It seems too tricky for a standard A level exam question as it stands.
I fed it to Wolfram, and it came out with a "simple" answer - admittedly I didn't check it was correct.
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atsruser
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(Original post by ghostwalker)
I fed it to Wolfram, and it came out with a "simple" answer - admittedly I didn't check it was correct.
I approached it more or less as you suggested. Maybe there's an easier method, but if so, I don't see it. Looks like a job for TeeEm.

How do you solve in this Wolfram?
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TeeEm
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(Original post by atsruser)
I approached it more or less as you suggested. Maybe there's an easier method, but if so, I don't see it. Looks like a job for TeeEm.

thanks for the compliment.

(Original post by _Caz_)
Hi I've just seen this come up on the Jun14 paper (fp3 ocr mei) and I can't find anything on the internet on how to do it. My teacher went through it in class and I still don't get it. The actual question was to find where the the plane

10x-y+2z=6

meets the surface

g(x,y,z)=x^2 + 3y^2 + 2z^2 + 2yz + 6xz - 4xy - 24 =0.

I understand that grad g has got to equal some multiple of the vector 10i-j+2k but don't know where to go from there.

Any help would be great thank you.

I cant find the mark scheme on the internet either or find any other questions like it.

Will try it now.
It will be more helpful if you post a picture
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_Caz_
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(Original post by TeeEm)
thanks for the compliment.




Will try it now.
It will be more helpful if you post a picture
Picture it's part (v)

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TeeEm
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(Original post by _Caz_)
Picture it's part (v)

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Just did it

I will upload in 3 minutes
not easy to explain or guide you.
the problem is a "given" that this is a tangent plane
I did it without given so I verified at the end.
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TeeEm
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(Original post by _Caz_)
Picture it's part (v)

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here it is
I hope it is correct.


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_Caz_
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(Original post by TeeEm)
here it is
I hope it is correct.


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Thank you I'm not sure what you did after it was in a matrix form tbh but it seems quite complicated to explain so you don't have to haha. The way I got shown in class was with a ton of simultaneous equations it was so confusing.

I suck at the whole module to be honest - I've kind of given up on it now. :mmm: thanks for doing the question though

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TeeEm
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(Original post by _Caz_)
Thank you I'm not sure what you did after it was in a matrix form tbh but it seems quite complicated to explain so you don't have to haha. The way I got shown in class was with a ton of simultaneous equations it was so confusing.

I suck at the whole module to be honest - I've kind of given up on it now. :mmm: thanks for doing the question though

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With the matrix I solved the simultaneous equations

It is known as the Jordan Gauss algorithm.

It is very useful and if you are going to study maths you will need it. Google it on U tube and watch a few examples.

good luck
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_Caz_
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(Original post by TeeEm)
With the matrix I solved the simultaneous equations

It is known as the Jordan Gauss algorithm.

It is very useful and if you are going to study maths you will need it. Google it on U tube and watch a few examples.

good luck
Thanks - I'm not doing a maths degree but it'll probably be useful somewhere down the line so I'll give it a look.


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ghostwalker
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(Original post by atsruser)
How do you solve in this Wolfram?
Just put both equations in, separated by a comma. May not be the correct method, and it produced the odd spurious output, but it did also give the same answer as TeeEm, with a little bit of work.
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TeeEm
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(Original post by _Caz_)
Thanks - I'm not doing a maths degree but it'll probably be useful somewhere down the line so I'll give it a look.


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you are not doing a maths degree but you are doing FP3! from MEI!?
The content of this unit is practically all undergrad, multivariable calculus, markov chains, differential geometry etc
WHY?
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_Caz_
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(Original post by TeeEm)
you are not doing a maths degree but you are doing FP3! from MEI!?
The content of this unit is practically all undergrad, multivariable calculus, markov chains, differential geometry etc
WHY?
Trust me its not by choice. I thought we'd be doing two applied modules and fp2. I'm (hopefully) starting a mechanical engineering degree in September which is why I picked further maths in the first place wasn't quite prepared for the sheer amount of weirdness that is FP3 haha.

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TeeEm
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(Original post by _Caz_)
Trust me its not by choice. I thought we'd be doing two applied modules and fp2. I'm (hopefully) starting a mechanical engineering degree in September which is why I picked further maths in the first place wasn't quite prepared for the sheer amount of weirdness that is FP3 haha.

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Then I am sorry to say that most of this FP3 will become very relevant in the first 2 years of a mechanical engineering course.

I hope everything goes well.
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_Caz_
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(Original post by TeeEm)
Then I am sorry to say that most of this FP3 will become very relevant in the first 2 years of a mechanical engineering course.

I hope everything goes well.
I suppose at least it's relevant for engineering. I'm a little worried now though because it is by far my worst module. Oh well at least I will be sort of familiar with it when it comes round to doing those topics in a degree.

Thank you for all of your help

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