Prove that if $V$ is an inner product space, then $|\langle x,y\rangle| = \|x\| \cdot \|y\|$ if and only if one of the vectors $x$ or $y$ is a multiple of the other.   Given the hint: identity holds and $y \neq 0$, let $a=\frac{\langle x,y\rangle}{\|y\|^2}$, and let $z=x-ay$. Prove $y$ and $z$ are orthogonal and $|a|=\frac{\|x\|}{\|y\|}$. Then use Pythagorean theorem   I am not able to appreciate this hint, can someone explain the general idea behind this? especially why do we need a $z$ a have to prove $z$ and $y$ are orthogonal? Want the intuition to understand why to do this.
Prove that if $V$ is an inner product space, then $|\langle x,y\rangle| = \Vert x \Vert\Vert y\Vert$ if and only if one of the vectors $x$ or $y$ is a multiple of the other.  I am given a hint to let $a= \frac{\langle x,y\rangle}{\|y\|^2}$, and let $z = x - ay$, then prove $y$ and $z$ are orthogonal and $|a|= \frac{\|x\|}{\|y\|}$. I don't know how they come up with this, what was the original idea?
Interesting tab on StackOverflow One day I directed to SO and was met with the awesome Interesting page, which I really like. But when I travel away from it, I have to re-enter the query string to get it back. Is there a plan to make this tab visible so I can click to it anytime? Show that the series $\sum\limits_{n=2}^{\infty} \frac {(n^3+1)^{1/3}-n}{\log n}$ converges Show that the series $$\sum\limits_{n=2}^{\infty} \frac {(n^3+1)^{1/3}-n}{\log n}$$ converges.   I showed it using Abel's theorem and limit comparison test. Any other simpler method? "A/An" preceding a parenthetical statement When a/an precedes a parenthetical aside (sometimes seen in informal/conversational writing), should the vowel rule depend on the first word in parentheses, or the next word in the "regular" flow of the sentence?     I need a (memorable) idiom (preceding an m word; use a)   or     I need an (memorable) idiom (preceding an i word; use an) $\sum k! = 1! +2! +3! + \cdots + n!$ ,is there a generic formula for this? I came across a question where I needed to find the sum of the factorials of the first $n$ numbers. So I was wondering if there is any generic formula for this? Like there is a generic formula for the series: $$ 1 + 2 + 3 + 4 + \cdots + n = \frac{n(n+1)}{2} $$ or $$ 1^{2} + 2^{2} + 3^{2} + 4^{2} + \cdots + n^{2} = \frac{n(n+1)(2n + 1)}{6} $$  Is there is any formula for: $$ 1! +2! +3! + 4! + \cdots + n! $$ and $$ {1!}^2 +{2!}^2 +{3!}^2 + \cdots + {n!}^2 $$?  Thanks in advance. If not, is there any research on making this type of formula? Because I am interested. How to use PSTricks in pdfLaTeX? I thought that PSTricks package was not possible to use in pdfLaTeX but .   How can I force pdfLaTeX to use PSTricks then? Why don't Muggle-born wizards use Muggle technology to fight Death Eaters? We know that wizards in the Harry Potter universe are subject to regular physical harm (e.g. witness injuries from being hit by Bludgers or just colliding with things when playing Quidditch).  Therefore, it's almost a certainty that an average wizard would be vulnerable to, say, a bunch of bullets.   There may likely be spell(s) to protect from projectiles - witness Dumbledore's  deployed when Voldemort sent a bunch of glass at him during the  at the end of Harry Potter and the Order of the Phoenix (according to , possibly an unusually powerful Protego or the Silver shield).   But:   those spells obviously take effort/energy/time to cast (not everyone's Dumbledore) AND more likely than not most Death Eaters wouldn't recognize a Muggle gun for a threat (or realize what kind of threat) till too late.   So, the guns should at best allow an over-matched OotP member (e.g. Harry or Hermione) to greatly equalize the power between any regular Death Eaters and themselves, and even perhaps help against Voldemort (who can't be killed with a bullet, but, at best, would need to expend magical energy and time into conjuring a shield, and, at worst, be stopped/interrupted/driven off).  So I'm looking for an explanation - ideally in-universe, but possibly just some statement by JKR - of why the Muggle-born OotP - who are likely very much at ease AND familiar with Muggle technology and live in the late 20th century - do not use guns. Or, for that matter, ANY Muggle technology of the late 20th century? Cue Arthur C. Clarke's "indistinguishable from magic" meme.  Even if they are on some kind of idiotic moral crusade to never kill their opponents (even the Jedi aren't this dumb), given the time-frame, non-lethal weapons would have already existed, such as rubber bullets and tasers.  NOTE: Please don't offer the "if they do it, Death Eaters would start using guns as well" theory. First, Death Eaters would have used the guns if they knew how to, anyway. Second, the guns, like any "secret weapon", could be reserved for strategic battles, say the defense of Hogwarts in Deathly Hallows.    The origin of the question stems from two things.  First, Harry's fascination with all things that are different in the magical world (e.g. moving photographs), based on his knowledge of the Muggle material world, as well as the somewhat amusing cluelessness of the "Muggle artifacts" expert, Mr. Weasley, regarding Muggle items and how they work.   Basically, from the books, it's clear that the Wizarding world wouldn't know details about guns or what they are for or how they work.  The second thing is it's clear, at least initially, that wizards are afraid of Muggles finding out about them. Hence all the efforts to cloak their society (basically, do you REALLY want every Muggle gunning for you, even with Magic)? The idea of how control over the Muggle world seems to be more indirect, by Death Eaters asserting mind-control over the government. This means there's an implicit understanding by Death Eaters that they don't stand a chance against armed Muggles even given the magical disparity.  Considering Harry and Hermione's overall adaptability and inventiveness (and brains in the latter case), it's mind-boggling that the idea of fighting Death Eaters with modern technology completely slipped their mind. I mean, it's a basic part of human myth (see Steel vs. Magic themes in Conan books, or Beowulf, or heck, A Yankee in King Arthur's Court). So it's not like the idea of using technology against magic would be impossible to come up with. The question begs itself - why not? Considering the fact that Rowling generally tried to be logical and consistent in building the HP universe, I feel like there must be some in-universe explanation for this that I just didn't notice.   (in-universe answers only unless there are some authoritative out-of-universe statements that are part of canon). The main effect will be non-significant if the interaction is significant? What is a socket? Book identification about boy who gets computer zapped into head How to send a header using a HTTP request through a cURL call? I wish to send a header to my Apache server on a Linux box.  How can I achieve this via a cURL call? How are questions in the 'hot' tab on a site selected? I know questions on the "hot" tab of a site homepage are based on some sort of views/velocity criteria, but does anyone know the actual algorithm used?  Background: I was considering a feature-request about the "hot" tab, but I don't have enough information to speak about it intelligently. markdown doesn't work properly Why do I get "Cannot redirect after HTTP headers have been sent" when I call Response.Redirect()? When I call Response.Redirect(someUrl) I get the following HttpException:     Cannot redirect after HTTP headers have been sent.   Why do I get this? And how can I fix this issue? How can I stop mobs from despawning? I am using Spigot, some plugins, and I want to make like a zoo on a map, with mobs behind glass cubes. The problem is that when I load my map, every time some of the mobs just vanish, empty cages. I put eggs again, they spawn, I go to my other map in the multiverse (/mv tp my_other_map) and when I come back, empty cages again, always the same ones...   I've  read it's a question of distance of me (distance in cubes unit), if I'm too far away, the mobs will despawn... Is there a way for them to stay? Convert HTML to PDF in .NET I want to generate a PDF by passing HTML contents to a function. I have made use of iTextSharp for this but it does not perform well when it encounters tables and the layout just gets messy.  Is there a better way? Bash: Syntax Error on @() Let $G$ be a finite group and suppose there exists $f\in\text{Aut}(G)$ such that $f^2=\text{id}_G$, i.e., $f$ is its own inverse, and such that $f$ has no fixed points other than the identity $e$ of $G$, i.e., $f(x)=x\Rightarrow x=e$. Show that $G$ is necessarily abelian.   While trying to do this exercise I noticed two facts.  First, $g$ and $f(g)$ have the same order because $o(f(g))|o(g)$ and, applying $f$ again and using $f^2=\text{id}_G$, $o(f(f(g)))=o(g)|o(f(g))$ and, once the order of any element is $\geq 1$, it follows that $o(g)=o(f(g))$. Also, it's easy to see that $g$ and $f(g)$ commute.  Second, there cannot exist such an automorphism if the order of $G$ is even, because $f(e)=e$ and we can form pairs like $\{g,f(g)\}$ with $f(g)\neq g$ that are invariant under $f$, i.e., $f(\{g,f(g)\})=\{g,f(g)\}$. But once the order of $G$ is even, proceeding with the construction of the pairs, we'll end up with just one element $\neq e$, so we must have some $\gamma\in G\setminus{e}$ with $f(\gamma)=\gamma$, contradicting the hypothesis.  Let $n$ be the order of $G$ and let's fix an enumeration of the elements of $G$, say $G=\{g_1,\ldots,g_n\}$. My approach was the following. For each $\sigma\in S_n$, let $x_{\sigma}=\prod_{i=1}^ng_{\sigma(i)}$. Then we have that $f(x_{\sigma})=x_{\sigma^{\prime}}$. If it's shown that $f(x_{\sigma})=e\;\forall\;\sigma\in S_n$, then we'd have $x_{\sigma}=e\;\forall\;\sigma\in S_n$, which implies that $G$ is abelian. The problem is that $S_n$ has $n!$ elements while $G$ has $n$ elements, so there repetitions among the $x_{\sigma}$, and we cannot apply directly the reasoning of the last paragraph.  Is this the right way, or is there an easier manner to solve this? Horizontal spacing issue in \listoftheorems using amsthm + thmtools After upgrading to 15.04, I have one additional driver (it did not appear in 14.10). What is the purpose of this driver and should I enable it? International TLD's vs. duplicate content I currently work at a pretty big website that has visitors from around the globe. My job is to help out on the SEO, and one thing we've been discussing lately is the use of international TLD's. The ones we use range between:   (partly) translated websites like .es and .de that serve most of the content in the country's language non-translated (english) websites for non-english languages (due to a lack of translations) like .ro and .cz english websites for english speaking countries with localized TLD's (.co.nz, .co.uk)   On one hand I really have the feeling this is causing a lot of duplicate content, especially for the last two categories of TLD's.   On the other hand though it seems a lot like country-specific TLD's tend to score a lot better in that country's Google. Would it be advisable to keep on using these domains, or should we canonicalize them all to the .com version?