Prove that the volume of the tetrahedron whose coterminus edges are $\vec{a},\vec{b},\vec{c}$ is given by $\frac{1}{6}\vec{a}.(\vec{b}\times\vec{c})$    I could not prove it.Please help me.
I drew a parallelepiped that is spanned by three vectors, and we know the volume is given by the absolute value of the determinant of the matrix - with the three vectors arranged in rows (or columns, since $detA = detA^T$).  How can I draw the tetrahedron from this parallelepiped to convince myself it is indeed 1/6th of the volume, i.e., 1/6 * |detA|?  Thanks,
What is the maximum value of $n$ if $n + 5$ divides $n^5 + 5$, $n$ being a natural number? I tried to solve by binomial theorem but it failed. Should flagging a question for closure prevent close-voting? How can I resolve the error "cannot execute binary file"? When I login using SSH, all I can see is this...  -bash: /usr/bin/id: cannot execute binary file -bash: [: : integer expression expected   I couldn't do anything in here. Commands such as halt, poweroff, reboot will return command not found.  How can I fix this? I am using Debian Squeeze Linux Is Ghost Adventures real or fake? How to show this equality of $\rho(X,Y)$ Let I have $(S,\Sigma,\mu)$ be a probability space, then $X,Y \in \Sigma$. Define $\rho (X,Y)$ by $\rho (X,Y)$ = correlation between random variable $I_X$ and $I_Y$, where $I_X$ and $I_Y$ are the indicator function of $X$ and $Y$. Express $\rho (X,Y)$ in term of $\mu (X)$, $\mu (Y)$, $\mu(XY)$. Conclude that $\rho(X,Y)=0$, if and only if $X$ and $Y$ are independent.      How to show that: $$\rho(X,Y)= \frac{\mu(XY)\,\mu(X^cY^c)-\mu(XY^c)\,\mu(X^cY)}{(\mu(X) \,\mu(X^c)-\mu(Y) \,\mu(Y^c))^{1/2}};0&lt;\mu(X)&lt;1;\,0&lt;\mu(Y)&lt;1$$ Why does the variance of the Random walk increase? consider the following subsets of complex plane $$\Omega_1=\left\{c\in\Bbb C:\begin{bmatrix}1&amp;c\\\bar c&amp;1\\ \end{bmatrix}\text{ is non-negative definite } \right\} $$  $$\Omega_2= \left\{c\in\Bbb C:         \begin{bmatrix}         1 &amp; c &amp; c \\         \bar c &amp; 1 &amp; c \\         \bar c &amp; \bar c &amp; 1 \\         \end{bmatrix} \text{ is non-negative definite } \right\}$$  Let $$\bar D=\{z\in\Bbb C:|z|\le1\}$$ Then 1) $\Omega_1=\bar D,\Omega_2=\bar D$  2). $\Omega_1\neq\bar D, \Omega_2=\bar D$  3). $\Omega_1=\bar D, \Omega_2\neq\bar D$  4). $\Omega_1\neq\bar D, \Omega_2\neq\bar D$  How to proceed? Thank you. Is past simple ever justified with 'by'? Can anyone shed some light on the following aspect?  By 5 PM I gave him the book.  By 5 PM I had given him the book.  In both the sentences the book was given before 5 PM and no later than 5 PM. But I can't understand then what the difference between them is. a Also is it right to use simple past with 'by' at all? If yes, then why and if not, then why is it incorrect to use simple past with 'by'? Angle notation for complex numbers in polar form I am looking for suggestions on how to typeset complex numbers in the modulo-argument form, sometimes called phasor notation. Have already checked  but nobody mentions this in particular. I am explicitly excluding exponential and sine-cosine notations. My MWE is as follows  \documentclass{article} \begin{document}     $z = r \angle \phi$ \end{document}   I think it looks ugly, specially if the angle has many digits, ie  $z = 1.19 \angle -78.2039^{\circ}$   I was thinking of an \angle replacement which maybe would extend the lower segment up to the last digit. What do you think?  EDIT: It looks like the steinmetz package doesn't do a good job with the vertical space if the \phasor{} argument has a \fraction{}{} with parenthesis, see:  \documentclass{article} \usepackage{steinmetz}  \begin{document}     Something.     \[I = I \phase{\left(\frac{A_0}{B_0}\right)}\]     This line is too close to the upper equation, don't you think?      \[I = I \phase{\left(\frac{A_0}{B_0}\right)}\]     Whereas this line is NOT so close to the upper equation. \end{document}   Strangely, commenting the first line (% Something.) fixes the problem. Is there any way to fix this? I already know how to do it on Mac OS X (plenty of instructions online).  But is it possible to do it on Ubuntu? If yes, how? 'Good' review audits can be misleading Explain how (or if) a box full of photons would weigh more due to massless photons I understand that mass-energy equivalence is often misinterpreted as saying that mass can be converted into energy and vice versa.  The reality is that energy is always manifested as mass in some form, but I struggle with some cases:  Understood Nuclear Decay Example  In the case of a simple nuclear reaction, for instance, the total system mass remains the same since the mass deficit (in rest masses) is accounted for in the greater relativistic masses of the products per $E=\Delta m c^2$.  When a neutron decays and you are left with a fast proton and a relativistic electron.  If you could weigh those two without slowing them down, you would find it weighed as much as the original neutron.  Light in a Box  This becomes more difficult for me when moving to massless particles like photons.  Photons can transmit energy from one heavy particle to another.  When a photon is absorbed the relativistic mass (not the rest mass) of the (previously stationary) particle that absorbs it increases.  But if my understanding is correct, the energy must still be manifested as mass somehow while the photon is in-flight, in spite of the fact that the photon does not have mass.  So let's consider a box with the interior entirely lined with perfect mirrors.  I have the tare weight of the box with no photons in it.  When photons are present the box has an additional quantifiable amount of energy (quantified below) due to the in-flight photons.  Say there are $N$ photons... obviously assume $N$ is large.  $$\Delta m = \frac{ E }{ c^2 } = \frac{ N h }{ \lambda c}$$  Interactions are limited to reflections with the wall, which manifest as a constant pressure on the walls.  If I hold this box in a constant gravitational field (like the surface of Earth) then there will be a gradient in the pressure that pushes down slightly.  Is this correct?  Wouldn't there still technically be mass as the photons are in-flight, which would cause its own gravitational field just as all matter does?  How is this all consistent with the assertion that photons are massless?  Is it really correct to say that photons don't have mass?  It seems to be a big stretch.  Please offer a more complete and physically accurate account of this mirror-box. I use a dual monitor.  The colors appear entirely different on the new monitor than they do on my laptop.  When you are selecting colors etc, to be included in a website design, it totally freaks me out.  Is there no software solution that finds how the settings of one of the monitor is and puts that to the second one?  Otherwise what all settings do I need to change manually on the second monitor to make it appear as it does on the laptop?  Thanks in advance. What are the differences between executing shell scripts using "source file.sh", "./file.sh", "sh file.sh", ". ./file.sh"? Have a look at the code:  #!/bin/bash read -p "Eneter 1 for UID and 2 for LOGNAME" choice if [ $choice -eq 1 ] then         read -p "Enter UID:  " uid         logname=`cat /etc/passwd | grep $uid | cut -f1 -d:` else         read -p "Enter Logname:  " logname fi not=`ps -au$logname | grep -c bash` echo  "The number of terminals opened by $logname are $not"   This code is used to find out the number of terminals opened by a user on the same PC. Now there are two users logged on, say x and y. I am currently logged in as y and there are 3 terminals open in user x. If I execute this code in y using different ways as mentioned above the results are :  $ ./file.sh The number of terminals opened by x are 3  $ bash file.sh The number of terminals opened by x are 5  $ sh file.sh The number of terminals opened by x are 3  $ source file.sh The number of terminals opened by x are 4  $ . ./file.sh The number of terminals opened by x are 4   Note: I passed 1 and uid 1000 to all these executables.  Now can you please explain the differences among all these? Today I taught an introductory class of statistics and a student came up to me with a question, which I rephrase here as: "Why is the standard deviation defined as sqrt of variance and not as the sqrt of sum of squares over N?"  We define population variance: $\sigma^2=\frac{1}{N}\sum{(x_i-\mu)^2}$   And standard deviation: $\sigma=\sqrt{\sigma^2}=\frac{1}{\sqrt{N}}\sqrt{\sum{(x_i-\mu)^2}}$.   The interpretation we may give to $\sigma$ is that it gives the average deviation of units in the population from the population mean of $X$.   However, in the definition of the s.d. we divide the sqrt of the sum of squares through $\sqrt{N}$. The question the student raises is why we do not divide the sqrt of the sume of squares by $N$ instead. Thus we come to competing formula: $$\sigma_{new}=\frac{1}{N}\sqrt{\sum{(x_i-\mu)^2}}.$$ The student argued that this formula looks more like an "average" deviation from the mean than when dividing through $\sqrt{N}$ as in $\sigma$.  I thought this question is not stupid. I would like to give an answer to the student that goes further than saying that the s.d. is defined as sqrt of the variance which is the average squared deviaton. Put differently, why should the student use the correct formula and not follow her idea?  This question relates to an older thread and answers provided . Answers there go in three directions:   $\sigma$ is the root-mean-squared (RMS) deviation, not the "typical" deviation from the mean (i.e., $\sigma_{new}$). Thus, it is defined differently. It has nice mathematical properties. Furthermore, the sqrt would bring back "units" to their original scale. However, this would also be the case for $\sigma_{new}$, which divides by $N$ instead.   Both of points 1 and 2 are arguments in favour of the s.d. as RMS, but I do not see an argument against the use of $\sigma_{new}$. What would be the good arguments to convince introductory level students of the use of the average RMS distance $\sigma$ from the mean? Present tense in Flashback story for something that is still true now? I'm wondering whether I should use the present tense for a descriptive statement but in a past context. For example:     (1) I went to the zoo and saw a giraffe last week.       (2) A giraffe is a mammal with long neck.       (3) It is the first time I see that animal with my own eyes.   Statement (2) is a description of a giraffe in general, not the specific one in the zoo, so is it correct to use is?  Statement (3) is an event that is still true now, so should I use is and see?  I'm not native speaker so forgive me if the question is very obvious to others. How to automatically archive a directory? I want to do one way synchronization.  I am having Folder A on my computer which is constantly updated with content. Another Folder B is used for backup purpose which is on external HDD.  Now what I expect is that whatever extra which is present in folder A should go to folder B. However something which is present in B and NOT in A ""shall NOT be copied to A"".  In a nutshell, the backup folder may copy everything from the source folder, however nothing should be copied form backup folder to the source. Is it possible to understand the One Piece storyline by watching the movies only? Introduction to statistics for mathematicians What is a good introduction to statistics for a mathematician who is already well-versed in probability?  I have two distinct motivations for asking, which may well lead to different suggestions:   I'd like to better understand the statistics motivation behind many problems considered by probabilists. I'd like to know how to better interpret the results of Monte Carlo simulations which I sometimes do to form mathematical conjectures.   I'm open to the possibility that the best way to go is not to look for something like "Statistics for Probabilists" and just go to a more introductory source. Can a set be neither open nor closed? An example would do. I cant think of any.   Thanks in advance!