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January 31

Normal vectors

For homework I have this problem: "You start at the point (1,0) and walk along the vector v = 4i + 2j. If you want to end up on the point (1,6) and only make 1 turn of 90 degrees, what are the coordinates of the turning point?" I know that the dot product of the two vectors must be 0, so I substituted y=(1/4)x+1 into vectors a and b to get $<x,\ {\frac {x}{4}}>$ · $<x-1,\ {\frac {x}{4}}-5>$ = (17/16)x² - (9/4)x. Solving for x I got (36/17,26/17) for the point to turn, which isn't correct. What am I doing wrong? 24.116.192.195 (talk) 00:37, 31 January 2010 (UTC)[reply]

I can't make any sense of what you've done. Where did y=(1/4)x+1 come from? You're starting at (1,0), then walking in the direction (4,2) for a while, then walking in the orthogonal direction (i.e. the direction (-2,4)) for a while. So you need to solve (1,6)=(1,0)+a(4,2)+b(-2,4), which has a unique solution. Algebraist 00:42, 31 January 2010 (UTC)[reply]

This seems a problem of unnecessary complication, introducing a vector for no good reason. If the wording was "You start ... walk along the straight line of slope 1/2 ...", you'd never think of a dot product and (I hope) get a solution in such a way as Algebraist has shown.→86.164.73.234 (talk) 01:07, 31 January 2010 (UTC)[reply]

Perhaps the student has not met the "product of gradients = -1" rule, and so needs to find a vector perpendicular to v = 4i + 2j using the scalar product. I agree though that Algebraist's method is much simpler than trying to use the vector equation of a line. Dbfirs 00:23, 2 February 2010 (UTC)[reply]

Rhombicosidodecahedron

Someone told me that you can't go round a rhombicosidodecahedron to visit every face once, but you can with any other Archimedean solid. Is it right? 4 T C 03:32, 31 January 2010 (UTC)[reply]

It's true there's not a way to do it for a rhombicosidodecahedron. If you look at the diagram with the colored faces in the article, the blue and red faces are only adjacent to yellow faces and vice versa, so every path has to alternate yellow and blue/red. But there are 32 blue/reds and only 30 yellows so that doesn't work. I don't know about the other Archimedean solids. Rckrone (talk) 07:09, 31 January 2010 (UTC)[reply]

Actually after a quick look it seems that icosidodecahedron, truncated dodecahedron, truncated cube and rhombicuboctahedron don't work either by the same argument. There may be others also. For each of these I assume you mean that you're only allowed to visit each face exactly once. Rckrone (talk) 07:13, 31 January 2010 (UTC)[reply]

Strange use of log

The last sentence of Hand sanitizer#Hand alcohol says:

Alcohol rub sanitizers containing 70% alcohol kill 3.5 log₁₀ (99.9%) of the bacteria on hands 30 seconds after application and 4 to 5 log₁₀ (99.99 to 99.999%) of the bacteria on hands 1 minute after application.

This is a very strange notation to me. It seems to be similar to nines (engineering). Is it actually in common usage in any field, or should it be changed to something more common? —Bkell (talk) 10:00, 31 January 2010 (UTC)[reply]

The reference quoted talked about a reduction of 3 log in bacteria meaning 99.9% killed or 0.1% remaining, it didn't put in the subscript 10 and I don't see that the 10 is needed. Dmcq (talk) 13:54, 31 January 2010 (UTC)[reply]

Correct or not, the notation is inappropriate for an article of that type. Encyclopedia articles are supposed to help people understand, not confuse them with arcane notation.--RDBury (talk) 14:55, 31 January 2010 (UTC)[reply]

I'm not so sure. I'd have thought wikipedia should be just reporting things as they are rather than trying to impose its own ideas of what measurements are allowable. Dmcq (talk) 16:41, 31 January 2010 (UTC)[reply]

... but we should not reproduce faulty notation. (I presume that the original document made more sense?) I see that the offending nonsense has now been removed. Dbfirs 09:43, 1 February 2010 (UTC)[reply]

Why not? It isn't Wikipedia's job to decide on correct notation. I'd have thought it should at least be documented in that article if it is a commonly used notation there, after all it measures what the article is all about. That's the sort of the thing the original document said, a reduction of 3 log in the bacteria. Dmcq (talk) 11:53, 1 February 2010 (UTC)[reply]

It depends on whether we view Hand sanitizer as primarily a medical article, or as primarily a non-technical "general interest" article. In the latter case, we should use the sort of standard English that we would see in a professional newspaper. In the former case, we should use the standard terminology from medical papers (whatever that is). If the medical literature usually says "a 5 log reduction" then there is no reason to avoid it here just because it sounds strange to mathematicians. — Carl (CBM · talk) 12:10, 1 February 2010 (UTC)[reply]

If we use a non-standard notation in a general article, we should explain its meaning, just as we should provide a translation if we quote a foreign language in an article written in English. The expression "a reduction of 3 log" has no meaning in mathematics or general science. Perhaps the medical article explains elsewhere that the phrase means a reduction of 3 on a (base 10) logarithmic scale graphing the number of bacteria? Dbfirs 00:17, 2 February 2010 (UTC)[reply]

Dmcq is correct in saying that medical sites use "3 log" in this context (though some use "3-log" to distinguish the usage from the ususal scientific meaning). (Personally, I would prefer to see "3_log" for this alternative meaning of "log", but this is unlikely to be taken up, so I'll forget it.) I've added a footnote explaining the medical notation. Dbfirs 20:08, 2 February 2010 (UTC)[reply]

Function harmonic on a strip

Hi all,

I've been trying to finish off this problem but I'm not quite sure where I'm going wrong - if anyone could give me any suggestions for how to get going, I'm happy to finish it all off myself, just need to know where I'm headed first!

This is the problem:

Let g(z) = exp(πz/a), h(z) = sin(πz/a) .

Show that g maps $\{x+iy:0<y<a\}$ onto $\{x+iy:y>0\}$

and h maps $\{x+iy:{\frac {-a}{2}}<x<{\frac {a}{2}},y>0\}$ onto $\{x+iy:y>0\}$ .

Find a conformal map of $\{x+iy:{\frac {-a}{2}}<x<{\frac {a}{2}},y>0\}$ onto $\{x+iy:0<y<a\}$ . (Done up to this point!) Find a function v which is harmonic on the strip −a/2 < x < a/2, y > 0 with limiting values on the boundaries given by: v = 0 on parts of the boundary in the left half plane (x < 0) and v = 1 on parts of the boundary in the right half plane. Is there only one such function?

Right, so the first few parts were fine, and I did what they obviously wanted and used the product $k(z)=g^{-1}(h(z))$ for the conformal map (product of conformal maps conformal etc) - now I know that if k(z) is conformal, it is also holomorphic, and the real part of any holomorphic function is harmonic, so then I tried to take the real part of $k(z)={\frac {a}{\pi }}\log(\sin({\frac {\pi z}{a}}))$ , where log is on the principal branch - I got something like (IIRC) ${\frac {a}{\pi }}\log({\sqrt {\sin ^{2}({\frac {\pi x}{a}})+\sinh ^{2}({\frac {\pi y}{a}})}})$ , which at the boundaries x=±a/2 is equal to ${\frac {a}{\pi }}\log({\sqrt {\cosh ^{2}({\frac {\pi y}{a}})}})$ , obviously not what we want - for one thing there's a y-dependence which if I understand the question correctly shouldn't be there. My only thought is that perhaps I shouldn't be taking the principal branch of log, since it looks like moving along the y=0 part of the boundary along the x-axis we have a discontinuity (0->1) at 0, and I can't see how else I might solve that part of the problem. Following that, how on earth would I be expected to confirm the uniqueness of such a function? I at least know what I'm meant to do, roughly, for the rest of the question, but when it comes to the uniqueness I'm totally stuck.

Thanks very much in advance, 82.6.96.22 (talk) 14:52, 31 January 2010 (UTC)[reply]

Try using arg instead of log (i.e. Im instead of Re of the log). As for uniqueness, I seem to remember there are a bunch of theorems on the uniqueness of harmonic functions, so maybe one of them applies here. Not sure though, need to think about it more.--RDBury (talk) 15:24, 31 January 2010 (UTC)[reply]

Actually Re(sin(πz/a) would be 0 on the boundary, so the answer to the uniqueness question is no.--RDBury (talk) 15:29, 31 January 2010 (UTC)[reply]

Is it? I can see why it would be 0 when x=±a/2, but what about when y=0? Surely we get a cosh term in y which will never be '0', and sin(pi x/a) is only 0 when x=±na, not ±a/2? Unless I'm missing something here... 82.6.96.22 (talk) 07:42, 1 February 2010 (UTC)[reply]

Intersecting cylinders

I've come across these results: the common volume of two cylinders of unit diameter whose axes intersect at right angles is 2/3, while that of three such cylinders whose axes intersect mutually at right angles is 2-√2. How can these be derived? The first one had a footnote saying that the result can be found without the use of calculus, rather suggesting that the second result cannot.→86.132.162.4 (talk) 15:08, 31 January 2010 (UTC)[reply]

There's a page about the shapes here - Steinmetz solid. In the first case how I'd do it is consider the sphere diameter 1 at the centre of the intersection. This has volume π/6, and by considering slices of the shape and sphere the ratio of the volumes is the ratio of the area of the circle and square. It should be possible to do the other shape in the same way, except in pieces, perhaps in three directions, which is where the sqrt 2 comes in, though it'd be a bit more work. --JohnBlackburne^words_deeds 15:19, 31 January 2010 (UTC)[reply]

The MathWorld page linked to from that article has derivations for both and gives formulas for a couple more complex intersections as well.--RDBury (talk) 15:38, 31 January 2010 (UTC)[reply]

Thanks - helped to know the name of the shapes.→86.132.162.4 (talk) 20:08, 31 January 2010 (UTC)[reply]

What is the link between the edges, faces and vertices of a 3D shape?

I already know what the faces, edges and vertices of a 3D shape are, but what do I need to do to the number of edges on any 3D shape (Except the cylinder) to get the same number as adding the number of faces and vertices together? Chevymontecarlo (talk) 16:16, 31 January 2010 (UTC)[reply]

Euler characteristic -- SGBailey (talk) 16:26, 31 January 2010 (UTC)[reply]

Thanks. Your link eventually led me to the answer. Chevymontecarlo (talk) 16:43, 31 January 2010 (UTC)[reply]