Wikipedia:Reference desk/Archives/Mathematics/2009 October 13

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October 13

Vectors and matrix algebra dot products et al

I have the X, Y, Z co-ordinates of 3 points in space for which I need to determine the following

1. The co-ordinate of a circles centre which passes through all 3 points

2. The co-ordinates of the point from 1. above, translated normal to the surface described by the original 3 points by a distance X.

It has been a very long time since I've used any vector or matrix so I'm a little muddled.

Any help would be great.

Solve first the special case where Z=0 for all three points. The general case is translated into the special case by changing coordinates. Bo Jacoby (talk) 07:03, 13 October 2009 (UTC).

... I don't think the OP was wanting the projection onto the XY plane? You have three vectors: OX, OY and OZ. The centre of the circle lies at the intersection of the perpendicular bisectors of the sides of the triangle XYZ, so you need to find vectors perpendicular to two sides (say XY and XZ) and passing through the middle of these sides, then find where these lines intersect. This is rather messy, so someone else will probably come up with a simpler method. For part 2, the normal to the plane of the circle is given by the vector product of any two vectors in the plane of the circle, such as XY and XZ (subtract co-ords to get these), find the vector product, divide by it's modulus, multiply by distance x, then add the result to answer to part 1. Dbfirs 07:35, 13 October 2009 (UTC)

... (later) ... If you are happy with a "magic formula" for the centre of the circle in part 1, it is given by the vector

$\alpha \,P_{1}+\beta \,P_{2}+\gamma \,P_{3}$

where

$\mathrm {P_{1}} ={\begin{bmatrix}x_{1}\\y_{1}\\z_{1}\end{bmatrix}},\mathrm {P_{2}} ={\begin{bmatrix}x_{2}\\y_{2}\\z_{2}\end{bmatrix}},\mathrm {P_{3}} ={\begin{bmatrix}x_{3}\\y_{3}\\z_{3}\end{bmatrix}}$

$\alpha ={\frac {\left|P_{2}-P_{3}\right|^{2}\left(P_{1}-P_{2}\right)\cdot \left(P_{1}-P_{3}\right)}{2\left|\left(P_{1}-P_{2}\right)\times \left(P_{2}-P_{3}\right)\right|^{2}}}$

$\beta ={\frac {\left|P_{1}-P_{3}\right|^{2}\left(P_{2}-P_{1}\right)\cdot \left(P_{2}-P_{3}\right)}{2\left|\left(P_{1}-P_{2}\right)\times \left(P_{2}-P_{3}\right)\right|^{2}}}$

$\gamma ={\frac {\left|P_{1}-P_{2}\right|^{2}\left(P_{3}-P_{1}\right)\cdot \left(P_{3}-P_{2}\right)}{2\left|\left(P_{1}-P_{2}\right)\times \left(P_{2}-P_{3}\right)\right|^{2}}}$

... but I recommend that you work through the long method to help your understanding of vector equations of lines. The vector equation of each perpendicular bisector is given by

b = OM + kp where O is the origin, M is the mid-point of a side, p is any vector perpendicular to the side, and k is an arbitrary constant (to be determined when you find where the lines intersect). Dbfirs 07:52, 13 October 2009 (UTC)

(edit conflict). I was suggesting to solve a special case, not a projection. The center (X,Y) of the circle has the same distance to the three points (X_A,Y_A), (X_B,Y_B), and (X_C,Y_C), so (X−X_A)² + (Y−Y_A)² = (X−X_B)² + (Y−Y_B)² = (X−X_C)² + (Y−Y_C)². Solving these two equations gives expressions for the two unknowns X and Y. First do the squares: X² − 2XX_A + X_A² + Y² − 2YY_A + Y_A² = X² − 2XX_B + X_B² + Y² − 2YY_B + Y_B² = X² − 2XX_C + X_C² + Y² − 2YY_C + Y_C². Then subtract X² + Y² giving − 2XX_A + X_A² − 2YY_A + Y_A² = − 2XX_B + X_B² − 2YY_B + Y_B² = − 2XX_C + X_C² − 2YY_C + Y_C². Now the equations are linear and the solution is straightforward. Bo Jacoby (talk) 08:39, 13 October 2009 (UTC).

Yes, I see. I suppose it is possible to extend this 2-D solution to 3-D, but the OP's heading suggested that they wanted a vector solution. Dbfirs 10:32, 13 October 2009 (UTC)

The center of the circle is a symmetric function of the 3 points. Your formula does not look symmetric. Can that be helped? Bo Jacoby (talk) 16:10, 13 October 2009 (UTC).

${(P_{1}-P_{2})\times (P_{2}-P_{3})=(P_{3}-P_{1})\times (P_{1}-P_{2})=(P_{2}-P_{3})\times (P_{3}-P_{1})=P_{1}\times P_{2}+P_{2}\times P_{3}+P_{3}\times P_{1}}$. — Emil J. 16:20, 13 October 2009 (UTC)

Thanks. I did make one error above, I should have said p is any vector perpendicular to the side in the plane of the triangle. I'm sure there should be a neater vector method for part 1, but I can't see it. My method is good for understanding vectors, but seems too long. There is a co-ordinate geometry formula for 3-D like Bo Jacoby's formula. I suppose I could derive it using my vector method if I had the patience, but it is a good exercise for the OP if he or she wants to understand vectors. Are there any experts who can see a short-cut? Dbfirs 16:44, 13 October 2009 (UTC)

An Almost-Primality Test

Is there any general way known to determine whether a number that fails a test for being prime is a product of only two primes without actually checking up to the number's third root for a factor? Is it just as efficient--so far as is known--to find a factor as to determine that it has at least three knowing that it has at least two? There are fast algorithms for determining whether a number is prime or not without actually finding a proper divisor. Once this is done with an affirmative (composite) result, I am left to find a factor in what may be a very long time when all I really want is the knowledge that there are or are not only two factors.Julzes (talk) 10:00, 13 October 2009 (UTC) For example, right now my computer is checking the factorization of 111110101011110010111110000010001₁₅₆₃ and it looks like it will take a week.Julzes (talk) 10:06, 13 October 2009 (UTC) I have a further question on this. I'm curious to know what the first two consecutive bases in which the above number is prime, so if anybody wants to run a search on that I would appreciate it. If it's likely that no such pair exists, I would like a good explanation also. My inclination is toward believing that such a pair probably does exist, but I haven't actually reasoned it out quantitatively.Julzes (talk) 10:10, 13 October 2009 (UTC) It is prime in both base 591 and 593, but aside from its being prime in base 2 and prime if we also allow it to be considered as in base 1, that's the closest I've come to what I'm looking for.Julzes (talk) 10:17, 13 October 2009 (UTC)

Would I be right to say that your number 111110101011110010111110000010001₁₅₆₃ is given by

1,610,516,746,593,217,788,093,321,222,359,707,265,648,849,509,955,413,660,168,610,383,205,686,470,714,500,112,576,660,497,611,201,902,703₁₀?

And if so, where did you get your desire to factorise such an exotic number? ~~ Dr Dec (Talk) ~~ 17:30, 13 October 2009 (UTC)

Without checking all 103 digits, it looks the same. I'm actually not so interested in the factorization as in knowing it has two prime factors and not three. I'm just following through on some research in great detail whose impetus will seem rather strange. It comes from reading the h2g2 Entry number of the article on the constellation Reticulum as a base-thirteen number. Got some rather nice arithmetical stuff from doubling the base conversion process. In this base-two case, the most interesting thing was the string of almost primes running from bases 29 through 33, along with the fact that before base 29 there are only the primes for the cases of 'base 1', base 2, base 7, and an almost-prime at base 16 (Every other smaller base has at least three prime factors). I've made an extensive set of listings for the other bases up to base 96, and I'm right now just in the process of extending the list for base 2.Julzes (talk) 23:06, 13 October 2009 (UTC)

The Dickman-de Bruijn function gives an indication of how likely a random number x is to have no prime factors above x^u x^1/u for given u. For u=3 it's about 5%. For a 103-digit number with 3 factors you could probably crank out a complete factorization with the elliptic curve method in a practical amount of time. 75.62.0.233 (talk) 23:41, 13 October 2009 (UTC)

What you just said doesn't quite make sense. I have no problem with the last sentence ('practical' is an irrelevant concept here anyway), but x obviously cannot have a factor greater than x³.Julzes (talk) 00:22, 14 October 2009 (UTC)

Sorry, misstated, see correction. Also this article. 01:16, 14 October 2009 (UTC)

Well, the number almost certainly has only two factors, but I'm still waiting.Julzes (talk) 09:16, 14 October 2009 (UTC) Still waiting. And now I've got my comp also simultaneously running for bases 1611 and 1813. It looks like I'm going to have at least two to add to the alpertron.com records list. There is something wrong with my Adobe Acrobat, so I haven't been able to open that article. I appreciate it, though. I guess it's unlikely anyone will come up with an almost-primality test any time soon, but perhaps there is a way to quicken things. I could probably declare the base-1563 case done, as it has no factor smaller than 35 digits (the borderline I need to consider), but I may as well wait it out and see just how large its smallest factor is.Julzes (talk) 04:31, 16 October 2009 (UTC)

Vectors in meterology

How are vectors used in meterology?Accdude92 (talk) (sign) 13:36, 13 October 2009 (UTC)

Wind velocity would be a vector (well, a vector field, strictly speaking - a vector for each point in the atmosphere). There will be other uses too, but that is the most obvious one. --Tango (talk) 16:40, 13 October 2009 (UTC)

• Any scalar field (e.g. temperature, air pressure, moisture, …) gives rise to a vector field; namely the gradient vector field. The gradient vector field has many useful physical interpetations. ~~ Dr Dec (Talk) ~~ 17:40, 13 October 2009 (UTC)

percent comparisons

Suppose I start with 100. Then 120 is 20% greater, 150 is 50%, 200 is 100%, and 300 is 200% greater. But 300 is also 3 times 100. So 200% greater is also 3 times greater. Is this seeming discrepancy merely a language artifact, or is there something deeper here? --Halcatalyst (talk) 19:13, 13 October 2009 (UTC)

"200% greater" is "3 times as great" and logically it would also be "2 times greater", but language isn't always logical and often uses "times greater" and "times as great" as synonyms (but only when no percent signs are involved); see this Q&A or this one, for instance. Certainly just a language quirk. —JAO • T • C 19:24, 13 October 2009 (UTC)

"200% of" is "2 times", "200% greater" is "3 times". I think "X times greater" is ambiguous and should be avoided. --Tango (talk) 19:32, 13 October 2009 (UTC)

Agreed, though it seems to be pretty much a lost cause. Even worse (to my ear) are such usages as "three times smaller" (meaning one-third as big, probably). AndrewWTaylor (talk) 09:54, 14 October 2009 (UTC)

I recall hearing "100% smaller" once, I have no idea what that was meant to mean (from context it didn't mean "reduced to zero", I think it was probably intended to mean "half", but why the speaker ever thought it did, I have no idea...). --Tango (talk) 10:23, 14 October 2009 (UTC)

Some probability questions...

1. I have a bag with 6 balls numbered one to six inside. What is the probability that I will pull the balls out in numerical order?

2. What is the probability if I do that again but this time I label the first ball as 1st (so ball marked "1" is correct) but placed sixth (so ball marked "6" is also correct); the second ball as 2nd (so ball marked "2" is correct) and placed fifth (so ball "5" is also correct)? And so on (3rd ball out must be either no.3 or number 4; 4th ball out must be either ball no.4 or no.3... etc to the end)

3. What is the probability for exercises 1 and 2 if I increase the number of balls to 7?

In case I am accused of asking a homework question, I'll tell you why I ask. I watch poker on TV and there is usually either 6 or 7 players at the table. Just for fun I try to predict the order in which people will be knocked out of the game. I used to do this by just writing down their placing. But lately I have been trying to "double" my chances (though I'm sure that isn't the truth in practice) by allowing myself to be correct using the two labels rather than just the one. I sense I may not have asked this question entirely clearly so please ask if you need me to clarify. --79.72.61.252 (talk) 23:38, 13 October 2009 (UTC)

For question 1, there are 720 permutations of the 6 balls and only one of them will satisfy the requirement, so the probability of getting that permutation (based on usual assumptions about drawing randomly) is 1/720. I don't understand question 2. In a given hand of poker, seat position has a real effect in one's winning chances--you can't assume independent draws. 75.62.0.233 (talk) 23:47, 13 October 2009 (UTC)

Thanks for your answer to question 1. That is much lower probability of success than I would have expected.

To clarify question 2, I am bringing out the numbers again but this time they can be in this order:

1st ball can be either ball 1 or 6

2nd ball can be either ball 2 or 5

3rd ball can be either ball 3 or 4

4th ball can be either ball 4 or 3

5th ball can be either ball 5 or 2

6th ball can be either ball 6 or 1

So given that there are now two correct balls at each stage what are my chances of succeeding at bringing them out in an order that is correct? —Preceding unsigned comment added by 79.72.61.252 (talk) 11:56, 14 October 2009 (UTC)

The chance of the 1st ball being right is 2/6=1/3. If that ball is right, the chance of the 2nd being right is 2/5. If that is right the chance for the 3rd is 2/4=1/2. After that only one of the options will be possible since the other will have already been used, so the chances for the 4th, 5th and 6th balls are 1/3, 1/2 and 1. To get the chance of them all being right we multiply those together. That gives us 2/180=1/90. That is quite a lot more likely than the 1/720 we had before (in fact, it is 8 times likely, that is 2³ since the first 3 balls are twice as likely to be right). --Tango (talk) 12:11, 14 October 2009 (UTC)

I don't think this is right. Once you know the value of the first ball, there is only one ball left out of the five remaining that can follow it. So we'd get the same probabilities as before, except for the first ball, which would be twice as likely to be correct. So the probablility of getting one of those sequences is 1/360 (or 2/720 - note there are 720 permutations and only two are correct). Readro (talk) 12:28, 14 October 2009 (UTC)

No, it would only be two permutations if the series had to be either 1,2,3,4,5,6 OR 6,5,4,3,2,1, however this allows for more permutations than that, such as 6,2,4,3,5,1 etc. --79.72.61.252 (talk) 14:08, 14 October 2009 (UTC)

So, therefore, if there were seven balls, the chance of pulling them out in the correct order is 1 in 7! = 5040. The chance of the numbers being correct according to the second system is 1 in 5040/8 = 630. More generally, where N is the number of balls, the odds are 1 in N! and 1 in N!/[2^INT(N/2)]. Warofdreams talk 16:03, 14 October 2009 (UTC)

Thanks everyone. --79.72.61.252 (talk) 11:40, 16 October 2009 (UTC)