Wikipedia:Reference desk/Archives/Mathematics/2013 June 5

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

June 5

Deriving the vector equation of the intersection of two planes

I'm taking a calculus class and I didn't catch the entire example of how to find the line at the intersection of two planes in three-dimensional space. Say I'm given these planes:

$\pi _{1}\equiv A_{1}\cdot x+B_{1}\cdot y+C_{1}\cdot z+D_{1}=0$

$\pi _{2}\equiv A_{2}\cdot x+B_{2}\cdot y+C_{2}\cdot z+D_{2}=0$

How do I determine the equation of the line in the form of ${\vec {r}}={\vec {r_{0}}}+t\cdot {\vec {d}}$? --Melab±1 ☎ 00:58, 5 June 2013 (UTC)

I have fixed your math code -- \\ does not work in math mode. Looie496 (talk) 01:32, 5 June 2013 (UTC)

Plane (geometry)#Line of intersection between two planes gives a fairly good explanation.--Salix (talk): 01:49, 5 June 2013 (UTC)

Convergence of an Infinite Series

I'm having trouble with a past exam question which is whether $\sum _{n=1}^{\infty }{\frac {\cos(nx)}{\sqrt {n!}}}$ converges? — Preceding unsigned comment added by 211.31.22.140 (talk) 10:01, 5 June 2013 (UTC)

If a sum is absolutely convergent, then it is convergent. Since $\left|\cos(nx)\right|\leq 1$ for all x and n, then by the limit comparison test your series converges if $\sum _{n=1}^{\infty }{\frac {1}{\sqrt {n!}}}$ does. Clearly, ${\frac {1}{\sqrt {n!}}}\leq {\frac {1}{2^{n}}}$ for large n, so by the same test, since $\sum _{n=1}^{\infty }{\frac {1}{2^{n}}}$ converges, your series does. Phoenixia1177 (talk) 10:44, 5 June 2013 (UTC)

Corrected notation: '\cos' instead of 'cos'. --CiaPan (talk) 05:51, 6 June 2013 (UTC)

Continuity and Limits

Thankyou! I was also having trouble with proving that if f and f' are continuous on the reals, that if they both have finite limits at infinity, this limits the limit of f' is zero and what the limit of f would be? — Preceding unsigned comment added by 211.31.22.140 (talk) 11:15, 5 June 2013 (UTC)

Consider the function $f(x)=a+{\frac {b}{x}}$. The limit of f is a, which can be chosen to anything. Bo Jacoby (talk) 11:38, 5 June 2013 (UTC).

Thankyou Bo, that answers half the question nicely :) But what about proving that the limit of f' is always 0? — Preceding unsigned comment added by 211.31.22.140 (talk) 12:09, 5 June 2013 (UTC)

Apply the mean value theorem to $\lim _{x\to \infty }(f(x+1)-f(x))$. On the one hand this limit is zero (since f has a limit at infinity) and on the other hand it's equal to the limit of $f'$ at infinity. Sławomir Biały (talk) 13:35, 5 June 2013 (UTC)

Can anybody help me ?

I posted a question last week about finding a relationship between elliptic integrals and the factorial/gamma function, to which I got no reply whatsoever. Did you guys miss my question by any chance, or is there truly nobody out there able to help me ? :-( — 79.113.220.191 (talk) 13:46, 5 June 2013 (UTC)

There's a paper by Borwein and Zucker on elliptic integral evaluation of special values of the gamma function. Apparently a number of such formulae go back to Gauss. Sławomir Biały (talk) 15:17, 5 June 2013 (UTC)

Resolved

Efficient way to divide

I'm trying to improve my calculation speed for simple arithmetic operations (such as 12×7=? and 348÷12=?) by practicing games such as this and this. Division is by far the most difficult operation, and I always get stuck when I come across a double digit divisor. e.g. when I start solving 3264÷68 I need to find ⌊320÷68⌋, but the only way I know to do that is to start guessing a digit (for example, 5), multiply (5×68=340) and adjust the answer (340>320 and therefor the quotient is less than 5) until I get it right. It's very time consuming and I wanted to know if there's a faster way to do it.

Thanks a lot, 79.182.171.46 (talk) 19:58, 5 June 2013 (UTC)

You could try approximation 326÷68≈320÷70=32÷7=4. That will give you a good first guess.--Salix (talk): 07:20, 6 June 2013 (UTC)

Whether one uses long division or short division or tries to do this in the mind, multi-digit divisors always involve a bit of guessing to get the right factors. Salix' method of approximating to one or two digits of precision is a good strategy, so is successive approximation. Eyeballing it, 3264 is about half of 68*100, so 50 is a a first guess. 68*50 = 6800/2 = 3400. How much less? We see that 3400-3264=136 which is twice 68. So the answer is 50-2 or 48. --Mark viking (talk) 11:18, 6 June 2013 (UTC)

There's a number of additional tricks one can do if you're sure the division will always be exact as in the example above. I did that one by dividing through both by 4 and then factorizing the first number so it came out pretty quickly without any writing down. Knowing the division is exact you know the end digit is 3 or 8 because they're the only ones that produce a result ending in 4 when multiplied by 8. Using 3 (which is wrong but no matter) I can take 3×68=204 from 3264 leaving 3060. I can then take 20 to leave 1700 and 17 goes into 68 4 times so I need to add 25 to give 48 altogether. Just a different direction for the division. Dmcq (talk) 11:29, 6 June 2013 (UTC)

With big divisions like this, remember that a division can be written as a fraction, and you know how to cancel fractions. 3264 ÷ 68 is the same as 3264 over 68 (I can't remember how to code this). So, you can divide the top and bottom of the fraction by the same amount, and you'll still have the exact same number, the exact same answer. In practice, this means 3264 ÷ 68 = 1632 ÷ 34 = 816 ÷ 17. In each of these steps, I have divided both numbers by two, but you could have done this in one step by dividing both numbers by four. Because I know 17 is a prime number, I know this is as far as I can go. You can do this simplifying trick with every division, whether or not the answer will be a whole number, whether or not it goes in nicely.

So now, I'm just seeing how many 17s go into 81, and at least that's easier to estimate. (17's a bit less than 20, and 4 x 20 is 80) 86.163.0.30 (talk) 11:52, 8 June 2013 (UTC)