I have always seen the formulas of rotation having cross product of $\vec{r}$ and some linear quantity. For e.g. $\vec{\tau} = \vec{r}$ x $\vec{F}$, $\vec{L} = \vec{r}$ x $\vec{p}$ etc. But we have never been told the source of these formulas. I never understood where they came from and what is their derivation. Is the direction of rotational quantities just a reference and accordingly, in consideration with magnitudes we have derived a cross product?  I have been pondering upon this question for half a year but I had no success.
If I'm not wrong, torque is perpendicular to both the radius and force i. e. It is along the axis of rotation. Questions that arise are- why do we consider the length between the axis/point of rotation while calculating torque? More importantly why is torque a cross product?
If $\sin \theta+\cos\theta+\tan\theta+\cot\theta+\sec\theta+\csc\theta=7$, then $\sin 2\theta$ is a root of $x^2 -44x +36=0$ My own bonafide attempt. $$ 0&lt;\theta&lt;\pi/2$$   and $$\sin\theta+\cos\theta+\tan\theta+\cot\theta+\sec\theta+\csc\theta=7$$   then show that $\sin 2\theta$ is a root of the equation $$x^2 -44x +36=0$$   I tried to use the above given equation of all the trigonometric ratios, though I ended up with an expression of $\sin\theta\cos\theta$.   But this too is in the form of $\sin\theta$ and $\cos\theta$ which was $\sin\theta\cos\theta=\frac{1}{6-\sin\theta-\cos\theta}$.  When I put that in the quadratic equation, it would again transform into the form of $\sin\theta$ and $\cos\theta$, and hence at last I couldn't prove the thing.  Even hints would work, as I would like to solve the question myself. Install Headers on Kali apt-get install linux-headers-`uname -r` Reading package lists... Done Building dependency tree Reading state information... Done E: Unable to locate package linux-headers-4.9.0-kali3-amd64 E: Couldn't find any package by glob 'linux-headers-4.9.0-kali3-amd64' E: Couldn't find any package by regex 'linux-headers-4.9.0-kali3-amd64' How to get images to planes objects to receive shadow? Whenever I try to Get then Install any app from the App Store, I get this message:     We could not complete your purchase.  cancelled   I did some searching online, but barely found any useful hits. What is this about?! Arrangements of MISSISSIPPI with all S's and P's separated If the all S,P are separated in the word MISSISSIPPI then the total possible arrangements are $k\frac {10!}{4!.4!} $ then value of k is? Now all S,P are separated so we can have slash method. Thus they can be placed alternatively in $6! $ ways and rest 5 can be placed in $5! $ ways. Thus we equate to get $$6!.5!=k\frac {10!}{4!.4!} $$  to get $k=13.7$ while $k=4/3$ .Where is my mistake Thanks proof $f(\overline{A})\subseteq \overline{f(A)} \Leftrightarrow f$ continuous How to duplicate a div in JavaScript ffmpeg - how to apply filters without losing quality Prove that $1 + \cos\alpha + \cos\beta + \cos\gamma = 0$ If $\alpha + \beta + \gamma = \pi $ and $\tan(\frac{-\alpha + \beta + \gamma}4)\tan(\frac{\alpha - \beta + \gamma}4)\tan(\frac{\alpha + \beta - \gamma}4) = 1$  Then prove that:     $1 + \cos\alpha + \cos\beta + \cos\gamma = 0$.   I have no idea how to go about this.   Please help. TikZ \foreach loop with macro-defined list I have a small problem with the TikZ \foreach loop whenever the list, over which we loop, is defined my a macro. The following code example will show my problem:  \documentclass{scrartcl} \usepackage[utf8]{inputenc} \usepackage{tikz}  \begin{document} Picture one:  \begin{tikzpicture} \foreach \x/\y in {1.0/2.0, 3.0/4.0} \node[draw] at (\x,\y) {\x--\y}; \end{tikzpicture}  Picture two:  \newcommand{\mymacro}{1.0/2.0, 3.0/4.0} \begin{tikzpicture} \foreach \x/\y in {\mymacro} \node[draw] at (\x,\y) {\x--\y}; \end{tikzpicture} \end{document}   The expected result is that both images should look the same. But in the second image, I only get one node - and both \x and \y are each time expanded to 1.0/2.0, 3.0/4.0. I hope someone can help. Bibtex style for "Statistics and Computing" I would like to create a bibtex style which answers the requirements of Statistics and Computing journal. How could I do that? Here is an example of how the bibliography should look like:  Journal article: Hamburger, C.: Quasimonotonicity, regularity and duality for nonlinear systems of partial differential equations. Ann. Mat. Pura Appl. 169, 321–354 (1995) Book: Geddes, K.O., Czapor, S.R., Labahn, G.: Algorithms for Computer Algebra. Kluwer, Boston (1992) Book chapter: Broy, M.: Software engineering — from auxiliary to key technologies. In: Broy, M., Denert, E. (eds.) Software Pioneers, pp. 10–13. Springer, Heidelberg (2002)  In the text, the references should be as follows:  Negotiation research spans many disciplines (Hamburger 1995).  Thank you for your help! My only problem is making them line up three-across and have equal spacing. Apparently, spans can not have width and divs (and spans with display:block) don't appear horizontally next to each other. Suggestions?  &lt;div style='width:30%; text-align:center; float:left; clear:both;'&gt; Is what I have now. How do I print all lines of a file with duplicate values in a certain column Given a file like this:  1,768,12,46576457,7898 1,123,435,134,146 2,345,6756856,12312,1311 5,234,567465,12341,1341 1,3245,4356345,2442,13 9,423,2342,121,463 9,989,342,121,1212   I would like to list all rows (in bash terminal) such that the value in column 1 appears at least twice (in column 1). The result should be  1,768,12,46576457,7898 1,123,435,134,146 1,3245,4356345,2442,13 9,423,2342,121,463 9,989,342,121,1212 Solution of limit $\lim\limits_{x\to 0} \frac {e^{-1/x^2}}{x} $ Small question, I'm trying to solve this limit but I just can't wrap my head around this problem.  $$\lim_{x\to 0} \frac {e^{-1/x^2}}{x} $$  L'Hopital just seems to make it messier.   It's probably pretty simple - I'd like to hear what I'm missing. Properties of $\liminf$ and $\limsup$ of sum of sequences: $\limsup s_n + \liminf t_n \leq \limsup (s_n + t_n) \leq \limsup s_n + \limsup t_n$ Let $\{s_n\}$ and $\{t_n\}$ be sequences.  I've noticed this inequality in a few analysis textbooks that I have come across, so I've started to think this can't be a typo:  $\limsup\limits_{n \rightarrow \infty} s_n + \liminf\limits_{n \rightarrow \infty} t_n \leq \limsup\limits_{n \rightarrow \infty} (s_n + t_n) \leq \limsup\limits_{n \rightarrow \infty} s_n + \limsup\limits_{n \rightarrow \infty} t_n$.  I understand the right two-thirds of the equation, and can prove it.  I'm stuck on the far-left third.  I don't understand how that is true. Find duplicate files Is it possible to find duplicate files on my disk which are bit to bit identical but have different file-names? Grid search vs. random search for hyperparameter tuning I am interested in comparing the computational cost of grid search vs. random search algorithm for hyperparameter tuning. Is random search computationally less expensive then grid search? Is there an article/paper that talks about this topic? Thank you, Prove that ${\left(\frac{x+1}{2}\right)}^{x+1} \leq x^x$ I want to prove that for each real number $x&gt;0$, the statement  $${\left(\frac{x+1 } {2}\right)}^{\!x+1} \leq x^x$$  is correct.   I’ve tried using mathematical induction methods, but haven’t found a solution. Although, using any method is appreciated. Do you have to want to hurt someone for the cruciatus curse to work? OK so the scene where Harry sees Voldemort making Draco tutur Rowel brings a few questions to my mind.      he felt the rage that did not belong to him posses his soul, saw a long room, only light by fire light, and the grate blond Death Eater on the floor,screaming and writhing, and a slighter figure standing over him, wand outstretched, while Harry spoke in a high, cold, merciless voice.   "More, Rowel, or shall we end it and feed you to Nagini? Lord Voldemort is not sure he will forgive you this time...You called me back for this, to tell me Harry Potter has escaped again? Draco, give Rowel another taste of our displeasure... do it, or feel, my wrath yourself!"   A log fell in the fire: flames reared, their light darting across a terrified, pointed white face-   I remember in the Order of the Phoenix Harry tries to do the cruciatus curse on Bellatrix, it doesn't work. She say this is because Harry doesn't mean it, he doesn't want to hurt he enough. If she's right about that (I'm sure she is) I don't see how Draco is managing to make to make Rowel writhe and scream, because it is clear he doesn't want to be doing it.   Also Harry was actually angry at Bellatrix and if his anger at her killing his godfather won't work... well you get my drift.   I know Draco is probably better at magic, but I feel Bellatrix is right, you really have to want them to hurt for it to work.  Next thing, Voldemort's having Draco do it why? Is he doing to upset Lucius, because that and lot of effort to go to, seeing as Voldemort is angry and he'd probably do a better job himself. Or is he trying to train himself a new soldier, I mean the other ones are getting old. It seems that the question has eventually become a series of questions....    An example goes as:     The 1st and (the) 2nd paragraphs of the article are extremely long.   Another example:     What are the situation, (the) task and (the) result of your story?   Is it necessary to use the in ()?