Can someone tell me the steps to solve this problem?     Prove by vector method that the point of intersection of diagonals of   trapezium lies on the line passing through the midpoint of parallel   sides.   I assumed the ratio of division of diagonals at intersection point as $a:1$ and $b:1$ and wrote the position vectors of the intersection point in terms of $a$ and $b$.But i'm getting too many variables.Any easier way out?
Prove,by vector method,that the point of intersection of the diagonals of the trapezium lies on the line passing through the mid-points of the parallel sides.    My Attempt: Let the trapezium be $OABC$ and that the O is a origin and the position vectors of $A,B,C$ be $\vec{a},\vec{b},\vec{c}$.Then the equation of $OB$ diagonal is $\vec{r}=\vec{0}+\lambda \vec{b}................(1)$ And equation of $AC$ diagonal is $\vec{r}=\vec{a}+\mu(\vec{c}-\vec{a}).......(2)$ And the equation of the line joining the mid points of $OA$ i.e.$\frac{\vec{a}}{2}$ and $BC$ i.e. $\frac{\vec{b}+\vec{c}}{2}$ is $\vec{r}=\frac{\vec{a}}{2}+t(\frac{\vec{b}+\vec{c}}{2}-\frac{\vec{a}}{2}).........(3)$. Here $\lambda,\mu,t$ are scalars. I do not know how to solve $(1)$ and $(2)$ and put into $(3)$ to prove the desired result. Please help me.Thanks.
Big-O Notation Proof Given the definition of Big-O, prove that $f(n) = n^2 - n$ is $O(n^2)$.  When I use the given definition, I get $n^2 - n \leq n^2 - n^2$ which means that $n^2 -n \leq 0$, which is not true. Is there some step I'm missing? The pronunciation of words which begins 'con' and 'com' Electromagnetic waves can be shielded by a perfect conductor. What about gravitational fields or waves? What tense should I use to describe something in the past but probably is still true? The title may be a bit confusing. See the below example:     The movie (was/is) so good.   The simplest guideline on deciding which tense to use, past tense or present tense, to describe an object comes down to whether the description was only true that that time or is true even to the present point of time.  But now we talking about literary work(movies,books,etc) of which quality tend to stay the same over time, which tense should we use? Differentiable but not continuously differentiable. Given $f: \mathbb{R} \rightarrow \mathbb{R}$ defined as $$f(x)=\left\{\begin{array}{cc}x^2\sin\left(\frac{1}{x}\right)&amp;,x\neq 0\\ 0&amp;,x=0\end{array}\right\}.$$ I am trying to prove $f$ is differentiable at $x=0$ but not continuously differentiable there. Inline elements shifting when made bold on hover I created a horizontal menu using a HTML lists and CSS. Everything works as it should except when you hover over the links. You see, I created a bold hover state for the links, and now the menu links shift because of the bold size difference.  I encounter the same problem as . However, the post does not have proper solution. I've searched everywhere for a solution and can't find one. Surely I can't be the only one trying to do this.  Does anyone have any ideas?  P.S: I don't know the width of the text in menu items so I cannot just set the width of the li items.    .nav { margin: 0; padding: 0; } .nav li {      list-style: none;      display: inline;      border-left: #ffffff 1px solid;  } .nav li a:link, .nav li a:visited {      text-decoration: none;      color: #000;      margin-left: 8px;      margin-right: 5px;  } .nav li a:hover{      text-decoration: none;      font-weight: bold;  } .nav li.first { border: none; } &lt;ul class="nav"&gt;     &lt;li class="first"&gt;&lt;a href="#"&gt;item 0&lt;/a&gt;&lt;/li&gt;     &lt;li&gt;&lt;a href="#"&gt;item 1&lt;/a&gt;&lt;/li&gt;     &lt;li&gt;&lt;a href="#"&gt;item 2&lt;/a&gt;&lt;/li&gt;     &lt;li&gt;&lt;a href="#"&gt;item 3&lt;/a&gt;&lt;/li&gt;     &lt;li&gt;&lt;a href="#"&gt;item 4&lt;/a&gt;&lt;/li&gt; &lt;/ul&gt; What's up with Vic? In the TV series Glitch, we see several people returning from the dead.  We see Vic, a police officer,      getting in a car accident and seemingly .   Afterwards, he behaves very strange. He searches for the doctor and the risen, whom he seems to know a lot about, sometimes even more than they know themselves.      He rekills Maria by breaking her neck.   And he threatens the other risen.  Most of the risen, especially those that have been dead longer, have some difficulty remembering their past lives. Vic on the other hand, seems to remember fine but to have a complete change of character instead.   Why is he after the other risen? How does he know so much about them? Why did he have such a change of character?     Yes, these are several questions, but they are closely related, so I opted to ask them in one question. How does conservation work in wormholes? My questions might seem naive as I haven't plunged into the mathematical descriptions, but wormholes are interesting even if I don't actually know them :)  First of all, is wormhole making two possible paths betwwen points (like the tunnel in most visualisations) or is it just making two points beside each other like in the game Portal? Of course, second case can be bent into a tunnel, but there would be nothing special about tunnel part and no preferred way to divide into "normal" and "tunnel" part of spacetime.  My main questions are these:   Is momentum conserved? Is kinetic energy conserved? How does potential energy work?   To elaborate... In the game Portal you can fall into floor and fly out of the wall or floor elsewhere with the same speed, but another direction (normal to the portal). So momentum is not conserved, but kinetic energy is. Is that how it would actually work? Where does the momentum go then?  And does it really allow you to increase your potential energy while retaining kinetic? It would seem fair to take away some kinetic energy if you are put in a region where you have more potential energy.  Either way is the change in energy non-continuous like in the game and discussion that's linked below? Or are there just more than one way to move through but the change in potential energy is smooth in any of those? That would mean more potential equilibrium points though...  Similar questions has been discussed , is it correct about how the falling guys gets a lot of energy (in form of a potential one) by draining it from the mouths? What is the proper way to handle data between scenes? I am developing my first 2D game in Unity and I have come across what seems an important question.  How do I handle data between scenes?  There seems to be different answers to this:   Someone mention using , while other people told me this should be used to store other things like screen brightness and so on. Someone told me that the best way was to make sure to write everything into a savegame everytime that I changed scenes, and to make sure that when the new scene loads, get the info from the savegame again. This seemed to me wasteful in performance. Was I wrong? The other solution, which is the one I have implemented so far is to have a global game object that isn't destroyed between scenes, handling all the data between scenes. So when the game starts, I load a Start Scene where this object is loaded. After this ends, it loads the first real game scene, usually a main menu.   This is my implementation:  using UnityEngine; using UnityEngine.UI; using System.Collections;  public class GameController : MonoBehaviour {      // Make global     public static GameController Instance {         get;         set;     }      void Awake () {         DontDestroyOnLoad (transform.gameObject);         Instance = this;     }      void Start() {         //Load first game scene (probably main menu)         Application.LoadLevel(2);     }      // Data persisted between scenes     public int exp = 0;     public int armor = 0;     public int weapon = 0;     //... }   This object can be handled on my other classes like this:  private GameController gameController = GameController.Instance;   While this has worked so far, it presents me with one big problem: If I want to load directly a scene, let's say for instance the final level of the game, I can't load it directly, since that scene does not contain this global game object.  Am I handling this problem the wrong way? Are there better practices for this kind of challenge? I would love to hear your opinions, thoughts and suggestions on this issue.  Thanks How do I run Command & Conquer Gold on Windows 7 64-bit (in XP mode)? I downloaded the first Command &amp; Conquer game  and burnt some CDs. I couldn't get the Setup.exe program to run in Windows 7 64-bit, as it must need some old DLLs. So I ran up the Windows XP Mode virtual machine, and was able to install it fine.   However, when I try to run the program, I just get an empty message box with an "OK" button. I've set the compatibility to Windows 95, and checked all of the options in the Compatibility tab, but it's not running. I also downloaded the patch suggested by EA, which is an updated version of the THIPX32.DLL file (copied into the C&amp;C installation folder).  Has anyone managed to get this working? I get it after updating in Synaptic Manager  I recently did a clean install of Ubuntu 17.04 from 16.10.  error message:-  W: Download is performed unsandboxed as root as file '/var/cache/apt/archives/partial/samba-libs_2%3a4.5.8+dfsg-0ubuntu0.17.04.1_i386.deb' couldn't be accessed by user '_apt'. - pkgAcquire::Run (13: Permission denied) Question is to prove :  $f : S^1 \to \mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$  I guess it would be helpful to use intermediate value theorem  Assuming $f(x)\neq f(-x)$ then given any $p\in (f(-x),f(x))$ (assuming $f(-x)&lt;f(x)$) there exists $y\in (-x, x)$ such that $f(y)=p$  I am not very sure of how to use this..   It would be helpful if someone can give some hint which would help me to solve this..  Thank you. After lethal first strike damage to an indestructible creature, will it still do damage? If I were to attack with a 4/4 indestructible creature and it's blocked by a 4/2 creature with first strike, would my indestructible creature still attack during the second combat damage step, or would it just tap, not assigning any damage? Determining whether two $k$-chromatic graphs are isomorphic (respecting vertex coloration) How do I achieve this kind of light star effect? I'm kinda new to Blender. Currently playing around with version 2.8 to practice some rendering. Is there a way to achieve this 'light-star' effect for the blooming objects with the emission? If there is an alternative way then it's fine too. See these images below for references. Thanks in advance: How to show that $C=C[0,1]$ is a Banach space Let $C=C[0,1]$ be the space of all continuous functions on $[0,1]$. Define $\|f \|=\max \ |f(x)|$. I want to show that $C$ is a Banach space.     Below is my attempt and I was wondering if it's ok.    I know I have to show that $C$ is a complete normed space. Clearly, $\|f\| \geqslant 0$ and $\|f\|=0 \Leftrightarrow f=0$. $\|cf \|=\max~|cf(x)|=|c|\max |f(x)|=|c| \cdot \|f\|$. $\|f+g\|=\max~|f(x)+g(x)|\leq \max~|f(x)|+\max~|g(x)|=\|f\|+ \|g\|$. So $C$ is a normed space.   Next, I show that every Cauchy sequence in $C$ is convergent. Let $\{f_n\}$ be a Cauchy sequence in $C$. Let $\varepsilon \gt 0.$ Then $\exists$ an $N_1$ such that $$ \max~|f_n(x)-f_m(x)| \lt \frac{\varepsilon}{2}$$ for $n, m \gt N_1$ and $x\in[0,1]$.  But there is a subsequence $f_{k_n} $, which converges to $f$. So $\exists$ an $N_2$ such that $$ \max~\left|f_{k_n} - f\right|\lt \frac{\varepsilon}{2}$$ for each $n\gt N_2$.  Now Let $N = \max\{N_1, N_2\}$, if $n \gt N$ then $k_n \geqslant n\gt N$. So we have $$ \max~\left|f_n(x) - f(x)\right| \leqslant \max~\left|f_n - f_{k_n}\right| + \max~\left| f_{k_n} - f\right| \lt\frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon.$$  Thus, $\|f_n-f\| \to 0$ as $n\to \infty$. $\quad \square$  Thanks. Rotation and translation of a conic-section I tried to solve the following exercises, so I want to ask you if my answers are correct.     1) Given the coordinates system $(O'; X'' Y'')$ asociated to the    basis $B=[{b_1=\frac{1}{\sqrt{2}}; \frac{1}{\sqrt{2}};  &gt; b_2=\frac{-1}{\sqrt{2}}; \frac{1}{\sqrt{2}}}]$ with origin in    $O'=(2;5)$ and the system $(O;XY)$ asociated to the standar basis, find the intersection between $Q: y''^2=-4 \sqrt{2} x''$    and the straight line in $(O,XY)$ $V=(0;5)+t(-1;1) t\epsilon R$    Well, first I tried to write $Q$ in terms of $X$ and $Y$, and not in terms of $X''$ and $Y''$. I used the formula:  $(\begin{matrix}  X''\\  Y'' \\  \end{matrix})$ =$(C)_B$ × $(\begin{matrix}  X-2\\  Y-5 \\  \end{matrix})$   Then I replaced those values in the equation of $Q$ and I got $Q: \frac{(-x+y-3)^2}{2}=-4 \sqrt{2} \frac{(x+y-7)}{\sqrt{2}}$   After that, I found the intersection between $Q$ and $V$ using equations and I got that the intersection points in $(O;XY)$ are $P_1(3;2)$ and $P_2(-1;6)$      2)Given the coordinates system $(O,X'Y')$ asociated to the basis    $B=[{b_1=\frac{1}{\sqrt{2}}; \frac{1}{\sqrt{2}};  &gt; b_2=\frac{-1}{\sqrt{2}}; \frac{1}{\sqrt{2}}}]$ with origin in    $O=(0;0)$ and the ellipse $9X'^2+5Y'^2-3X'+2Y'-10=0$ find the    coordinates of the centre of the ellipse in $(O,XY)$    So, here I found the centre of the ellipse in $(O,X'Y')$ and it was $(\frac{1}{6}; - \frac{1}{5})$, then, doing something similar to what I've done before I found out that the centre in $(O,XY)$ was $( \frac{11}{30 \sqrt{2}};$ $\frac{-1}{30 \sqrt{2}}$) Do there exist general conditions under which we can conclude that continuity on a topological space is detected by $\mathbb{R}$? Whenever $X$ is a topological space, let us say that continuity on $X$ is detected by $\mathbb{R}$ iff for all functions $f : X \rightarrow Y$ where $Y$ is another topological space, we have that if every continuous function $c : \mathbb{R} \rightarrow X$ satisfies "$f \circ c$ is continuous", then $f$ is itself continuous.   Clearly, continuity on $\mathbb{R}$ is detected by $\mathbb{R}$ It is shown  that continuity on $\mathbb{R}^2$ is detected by $\mathbb{R}$. Clearly, it is not the case that continuity on $\mathbb{Q}$ is detected by $\mathbb{R}$, since every continuous mapping $\mathbb{R} \rightarrow \mathbb{Q}$ is constant.      Question. Do there exist general conditions underwhich we can conclude that continuity on a topological space is detected by $\mathbb{R}$? Scanner issue when using nextLine after nextXXX If just a single photon hit a single slit interferometer, what would happen? Would you just see a dot on the screen, or would there be a diffraction pattern? Furthermore, if you had a double slit interferometer but also had which-path information would there still be a diffraction pattern even though there is no interference?