Wikipedia:Reference desk/Archives/Mathematics/2015 March 31

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March 31

Trisecting the angle-there must be something wrong with this

Excuse me. I really don't know where to put this. I don't know what's wrong with this. If you can answer me, please remove this and PM me. In angle <AOC, If you used a compass to draw a circle, intersecting with AO and OC at a B and D, so that BO=DO then connected the intersections creating line segment BC. If you split that into three parts, BM, MN and NC(Agh... Trisecting the line should be self explanatory. I'm not good at explaining :D) Because OD=OB, <OBD=ODB (Isoceles triangle) and BM=NC, triangleOBM=triangleODN so <BOM=<DON. Furthermore, MN=BM=DN, and OM=OM, ON=ON so triangleOBM=triangleODN=triangleOMN, so <BOM=<MON=<DON. Please reply :p I know trisecting the angle is impossible, and if this were the proof, it would have been discovered long ago. As far as I'm aware, this is purely compass and straightedge. Can sombeody help? :D Someone with a Question (talk) 14:28, 31 March 2015 (UTC)

Just to let you know, I noticed you posted this at Angle trisection, I moved it to the talkpage Talk:Angle trisection, as that is a correct place to ask. Here is also a good place to ask. Unfortunately I don't actually know the answer, but good luck finding out. Joseph2302 (talk) 14:32, 31 March 2015 (UTC)

Your construction is very confusing and I think there are several issues with it.

One thing I noticed: You say OD=BD, this seems to make no sense given the way D was constructed. -- Meni Rosenfeld (talk) 15:03, 31 March 2015 (UTC)

Sorry typo. Fixed.

Your wording is confusing. A diagram would help. --David Biddulph (talk) 15:21, 31 March 2015 (UTC)

After your recent change it even more clearly doesn't make sense. If B and D are the intersections of a circle (centred at O) with OA and OC respectively, BD cannot be pependicular to BC. As I said, draw a diagram and post that. --David Biddulph (talk) 15:33, 31 March 2015 (UTC)

Um.. how? Oh. http://tinypic.com?ref=2mhx6p3 Someone with a Question (talk) 15:38, 31 March 2015 (UTC)

I can't see any picture there, but your words (after your latest amendment) make no sense at all. There are so many errors and inconsistencies that I wouldn't know where to start. If you want to try uploading a diagram, try loading it to Commons:, but please look carefully at your diagram and compare it with the words you have tried to use then you will hopefully see for yourself where you've gone wrong. --David Biddulph (talk) 16:00, 31 March 2015 (UTC)

[Misplaced question about dot and cross products removed by BenRG] Um.. I'm only in eighth grade. Is there any way to use easier terms? :D Sorry Someone with a Question (talk)

I'd just answered at the talk page of the article, but it is easy to see the construction doesn't work if you start off with a large angle making almost a straight line at O. The side angles are small and the middle one is most of the original. Dmcq (talk) 15:48, 31 March 2015 (UTC)

Oh... wait, I see. I found the mistake ^^. Thanks. Sorry for troubling you. But does anyone know what the guy with Lie Algebra is talking about?Someone with a Question (talk) — Preceding undated comment added 15:57, 31 March 2015 (UTC)

• @Someone with a Question: If I am reading (and part guessing) right, your construction rests upon the presumption that "In an isosceles triangle, the angle trisector of the vertex angle, trisects the base". However that is not true, so any construction based on this "principle" would fail. Abecedare (talk) 16:05, 31 March 2015 (UTC)

The Lie algebra guy was trying to ask a new question, not respond to yours. -- BenRG (talk) 17:23, 31 March 2015 (UTC)

I had no trouble understanding this and I'm surprised that everyone else was so confusedas it's currently written. There are a number of places where you wrote C when you meant D: "creating line segment BC" should be "creating line segment BD", and all later mentions of C should also be D. Aside from that, everything looks right until "triangleOBM=triangleODN=triangleOMN". The first two triangles are congruent but the third isn't. It looks like you were going for a SSS congruence but matched two of the sides of OMN to the same side of OBM and ODN. -- BenRG (talk) 17:21, 31 March 2015 (UTC)

Ben - Have you seen the original wording, or the current, heavily modified one? The original wording was much more confusing and convoluted, with many errors.

It was clear that he erroneously tried to prove that trisecting a segment trisects the angle, but following his convoluted construction to disprove his claim was difficult. -- Meni Rosenfeld (talk) 19:00, 31 March 2015 (UTC)

You're right, sorry. -- BenRG (talk) 21:49, 1 April 2015 (UTC)

Motivating dot and cross products

The vector analysis texts start with the arbtitrary definition of cross product A×B = |A||B| sinθ n.Similarly for dot product I can see a definition.However there is no mention of justification or proof or guidance as to how to intuitively arrive at this definition.What is the reason behind these definitions and why this obfuscation. Did the early prodigies Newton,Liebniz,Euler,Bernoulli or Gauss had know these two concepts.What is the reasoning for this definition before resorting to Lie Algebra because that came much later and was developed much later if not I am wrong — Preceding unsigned comment added by 103.24.110.221 (talk) 15:18, 31 March 2015 (UTC)

• (I took the liberty of reformatting the formula with some characters that you may not know how to make; I hope you don't mind.)

What obfuscation?

For some motivations, see Cross product#Applications and Dot product#Physics (though the latter could use some expansion).

Clearly if it's called a product, its magnitude must be proportional to those of each of the 'factors'. That explains part of the definitions.

Various physical applications call for a vector orthogonal to the two input vectors; that explains n. These same applications want the cross product to be zero when the input vectors are parallel (partly because otherwise "orthogonal to both" is under-determined), and of maximum magnitude when they're orthogonal. Thus sinθ.

The dot product |A||B|cosθ measures how similar two vectors are.

I make heavy use of both products in designing objects for 3D printing. I don't know nothin bout no Lie algebra. —Tamfang (talk) 07:24, 1 April 2015 (UTC)

Two practical problems that lead fairly naturally to dot and cross products are finding the angle between two vectors and finding a normal to the plane spanned by two vectors. Finding the angle between vectors is accomplished by the Law of cosines, the vector form of which is 2|u||v|cosθ = |u|² + |v|² - |u-v|². Divide this by 2 to get two alternate definitions of the dot product. If you work out the equation for the plane spanned by two given vectors, and clear fractions, the coefficients of x, y, and z are coordinates of the cross product (up to sign since the definition depends on orientation). It's possible that a book on vector analysis would assume that the reader is already familiar with linear algebra and geometry, and so would not need motivation for the more basic definitions. --RDBury (talk) 07:57, 1 April 2015 (UTC)

As a side note, ℝ³ with the cross product (which is anti-symmetric and fulfills the Jacobi identities) is a Lie algebra, see also here. YohanN7 (talk) 12:47, 5 April 2015 (UTC)