- Forums
###### Further math vecotr question

watch this thread

3 weeks ago

Hi!

I'm stuck on a vectors question on converting different forms of the same plane equation

Any help would be amazing!

I'm stuck on a vectors question on converting different forms of the same plane equation

Any help would be amazing!

Reply 1

3 weeks ago

It's going from

r=(3, 1,1)+a(1,3,-1)+b(2,5,0)

to r. (ci,d j, ek) = f

r=(3, 1,1)+a(1,3,-1)+b(2,5,0)

to r. (ci,d j, ek) = f

Original post by MayaVellichor

It's going from

r=(3, 1,1)+a(1,3,-1)+b(2,5,0)

to r. (ci,d j, ek) = f

r=(3, 1,1)+a(1,3,-1)+b(2,5,0)

to r. (ci,d j, ek) = f

What do you understand about the different forms?

Reply 3

3 weeks ago

Original post by mqb2766

What do you understand about the different forms?

I know that (ci, d j, ek) is the normal to the plane

And that r.n=a.n where r is general and a is a specific point on plane (I'm assuming that is the (3,1,1))

But I don't know how to find the normal from it

Original post by MayaVellichor

I know that (ci, d j, ek) is the normal to the plane

And that r.n=a.n where r is general and a is a specific point on plane (I'm assuming that is the (3,1,1))

But I don't know how to find the normal from it

And that r.n=a.n where r is general and a is a specific point on plane (I'm assuming that is the (3,1,1))

But I don't know how to find the normal from it

For the first form, it says that any point on the plane can be formed from a linear combination of the two direction vectors and adding that to the point you mention. So how are the two direction vectors and the plane normal related? A sketch may help (as usual).

(edited 3 weeks ago)

Reply 5

3 weeks ago

Original post by mqb2766

For the first form, it says that any point on the plane can be formed from a linear combination of the two direction vectors and adding that to the point you mention. So how are the two direction vectors and the plane normal related? A sketch may help (as usual).

are they directions along the plan and therefore the dot product of...

Ooooh the dot product of each vector with the normal will equal 0 cause of cos90? So then I solve simultaneously to find the normal?

Original post by MayaVellichor

are they directions along the plan and therefore the dot product of...

Ooooh the dot product of each vector with the normal will equal 0 cause of cos90? So then I solve simultaneously to find the normal?

Ooooh the dot product of each vector with the normal will equal 0 cause of cos90? So then I solve simultaneously to find the normal?

Yup, or use the cross product to get it directly (if youve covered it). Then once you have the normal, you can use the point to get d as per the previous post.

Original post by MayaVellichor

are they directions along the plan and therefore the dot product of...

Ooooh the dot product of each vector with the normal will equal 0 cause of cos90? So then I solve simultaneously to find the normal?

Ooooh the dot product of each vector with the normal will equal 0 cause of cos90? So then I solve simultaneously to find the normal?

Its also worth noting that even if you didnt understand the geometry insight, could sub equation 1 into equation 2. Then when you take the dot product with each termm, the left hand side would depend on a and b whereas the right hand side wouldnt. The only way this could occur for all a and b (free parameters) would be if the dot product of the corresponding direction vectors with the normal is zero in both cases.

While it falls out of the algebra, its "better" to understand the properties (geometry).

Reply 8

3 weeks ago

Original post by mqb2766

Its also worth noting that even if you didnt understand the geometry insight, could sub equation 1 into equation 2. Then when you take the dot product with each termm, the left hand side would depend on a and b whereas the right hand side wouldnt. The only way this could occur for all a and b (free parameters) would be if the dot product of the corresponding direction vectors with the normal is zero in both cases.

While it falls out of the algebra, its "better" to understand the properties (geometry).

While it falls out of the algebra, its "better" to understand the properties (geometry).

Yes I tried that where the normal is 0 but I don't entirely know what that means

Cause every plane should have a normal right?

Reply 9

3 weeks ago

Or is it more that they can take many values but they have to be in a certain ratio

Original post by MayaVellichor

Yes I tried that where the normal is 0 but I don't entirely know what that means

Cause every plane should have a normal right?

Cause every plane should have a normal right?

Not too sure what you mean, but if you have

n.(p + ar + bs) = d

where n, p, r and s are vectors as above, then the normal vector n must be non-zero and indeed its not unique, so any non-zero multiple of it is a normal vector as well. You can use the same representation of a line in 2d so

y = mx + c

-mx + y = c

(-m,1).(x,y) = c

where the normal is (-m,1) and it has a gradient -1/m as youd expect and d=c. Obv, a line is 1d in 2d space so a normal must exist, just as a plane is 2d in a 3d space, so again the a normal must exist. Its not unique as the same line is given by

(-2m,2).(x,y) = 2c

for instance. Its an equation, you can multiply through by a non-zero value.

So expanding you have

n.p + a n.r + b n.s = d

and the only way this can occur for all a and b is if n.r=n.s=0, so n is orhogonal to r and s. Then d = n.p

(edited 3 weeks ago)

- Quick moments Q
- I have a lot OF A-level and GCSE Maths and Further Maths resources. Who's Interested?
- Further maths
- A Level Further Maths
- Struggling with Further Maths
- gcse further maths
- Is Further maths, Maths, and Physics A-level a good combination?
- Which A level is harder physics or further maths?
- UCAS Clearing 2023 FAQ and Help Megathread
- Got a Ucas or results related question? Ask a Clearing and Applications Advisor here
- GCSE Further Maths
- maths help
- Gcse and A level results
- A Level options for Natural Sciences
- GCSE Mathematics Study Group 2023-2024
- Really not sure as to what A-level to drop, if Mechanical Engineering is my goal.
- 4th a level
- Sport science and mathematics joint honours
- AABU clearing?
- Hardest topics in Further Maths

- GCSE Mathematics Study Group 2023-2024
- Hypothesis Testing
- Step support programme assignment 7 help pls
- Senior Maths Challenge 2023
- A-level Mathematics Study Group 2023-2024
- Dimensional Analysis
- Do we have to memorise all physics equations for GCSE 2024 ?
- Girls Maths Olympiad 2023
- Electrical Engineering maths help with Desmos graphing calculator
- A Level Maths Moments question
- Discrete maths questions
- Plotting data on a graph.
- Hypothesis Testing
- Problem solving the smc 2023
- Maths trig
- Help With A level Trig
- which calculator; the fx-991cw or fx-991ex
- AS Maths edexcel 2023
- Edexcel AS Stats and mechanics paper 2023?
- Edexcel A Level Mathematics Paper 2 unofficial mark scheme correct me if wrong

- The Official Cambridge Applicants for 2024 Entry Thread
- ATAS certificate for UK for January 2024 intake
- University of Oxford 2024 Undergraduate Applicants Official Thread
- Any recommendations for online courses?
- Good A-level grades and bad gcse grades
- Advice for applying to optometry
- Personal Statements
- Official: Kent and Medway Medical School A100 2024 entry
- I’m taking Biology A level, and love animals, what degrees can I do?
- Official: University of Cardiff A100 2024 Entry Applicants

- 2024 Pharmacy applicants mega-thread
- glasgow offer
- Dwp work coach september 2023 merseyside
- Official: Barts and the London (QMUL) A100 2024 Entry Applicants
- How much is Second year of Uni worth?
- degree apprenticeship cv
- UCAS Letter of Reference -- employer?
- University of Lincoln A10L (Nottingham at Lincoln) Applicant thread 2024
- from all 9s to all Ds, help
- Manchester uni or KCL?

- Official: University of East Anglia (UEA) A100 2024 Entry
- Official: University of Leeds A100 2024 Entry
- New Thread Experience Feedback - Official Thread
- Official Thread: (Undergraduate) Medicine 2024 Entry
- Royall Holloway
- Sevenoaks School sixth form entrance
- Exam Board Hurt or Heal Game
- FANB1984's GCSE (YR 11) GYG
- Oxford College hurt/heal
- British food hurt/heal game!

- Official Imperial College London Offer Holders Thread for 2024 entry
- Breakfast items hurt/heal game!
- The Cambridge College Hurt/Heal Game [part 2]
- University of Manchester vs Leeds vs Sussex
- [Official thread] Hamas-Israel Conflict
- Agent Duck’s GCSE journey - GYG
- Any tips for cycling?
- Oxford Biochemistry Interview 2024
- Thoughts on the current situation between Palestine and Israel?
- Architecture Applicants 2024

The Student Room and The Uni Guide are both part of The Student Room Group.

© Copyright The Student Room 2023 all rights reserved

The Student Room and The Uni Guide are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: Imperial House, 2nd Floor, 40-42 Queens Road, Brighton, East Sussex, BN1 3XB