find the size of a triangle using (determinant) with the following points:    $(x_1,y_1)=(1,-2)$    $(x_2,y_2)=(-4,-2)$   $(x_3,y_3)=(-5,-1)$      How should I place those points in the determinant? or firstly in a matrix?
I'm not sure how to solve this problem. Can you guys provide some input/hints?      Let $A=(x_1,y_1)$, $B=(x_2,y_2)$ and $C=(x_3,y_3)$ be three points in $\mathbb{R}^{2}$.      Show that the area of $\Delta ABC$ is given by   $\frac{1}{2}\left| \det(M) \right|$, where   $$M = \begin{pmatrix} 1 &amp; 1  &amp; 1\\  x_1 &amp; x_2 &amp; x_3\\  y_1 &amp; y_2 &amp; y_3 \end{pmatrix}$$
How can I do a full shutdown of windows 10 without logging in? Sunlight Sensitivity looks pretty bad — Disadvantage to attack rolls and Perception checks whenever you, your target, or what you're trying to perceive are in direct sunlight. I hope I'm missing something, since no other PC race has a comparable hindrance.  Are there any ways you have used to mitigate this flaw to make players of this race not at a severe disadvantage (or, if it doesn't need mitigation, explain why it's already balanced without any action required)? I'm trying to prove the problem below - which comes from Munkres' "Analysis on Manifolds" book in the section on the Inverse function theorem.  Since its in the chapter on the Inverse Function Theorem I figured I'd start by showing that $f$ satisfies the conditions of the theorem.  Writing the Jacobian shows that it's both $C^r$ and we get $\det f'(x,y)=4x^2+4y^2\neq0$ when $x&gt;0$.  So we can apply the theorem but I'm unsure of how to proceed to show that $f$ is $1$-$1$.  And I didn't see how to use the hint they provided.  Thanks!      Let $f\colon \mathbf R^2\to \mathbf R^2$ be defined by the equation   $$f(x,y)=(x^2-y^2,2xy).$$   (a) Show that $f$ is one-to-one on the set of all $(x,y)$ with $x&gt;0$. [Hint: If $f(x,y)=f(a,b)$, then $\|f(x,y)\|=\|f(a,b)\|$.] Run command whenever someone joins a Minecraft server Is there any way to run a command automatically when a player joins the server? I need to do this without the use of a Bukkit plugin. I think it's possible with MCP but I want to know how! Finite Sum of Power? When is unbalanced data really a problem in Machine Learning? We already had multiple questions about unbalanced data when using , , ,  and a number of other similar questions, what makes it a very popular topic! Unfortunately, each of the questions seems to be algorithm-specific and I didn't find any general guidelines for dealing with unbalanced data.  Quoting , dealing with unbalanced data     (...) heavily depends on the learning method. Most general purpose   approaches have one (or several) ways to deal with this.   But when exactly should we worry about unbalanced data? Which algorithms are mostly affected by it and which are able to deal with it? Which algorithms would need us to balance the data? I am aware that discussing each of the algorithms would be impossible on Q&amp;A site like this, I am rather looking for a general guidelines on when it could be a problem. Partition on sublists beginning with a certain marker elements Working with large images and transform in the VSE How to evaluate $\int_{0}^{+\infty}\exp(-ax^2-\frac b{x^2})\,dx$ for $a,b>0$ I would like to change a file extension from *.txt to *.text. I tried using the basename command, but I'm having trouble on changing more than one file.  Here's my code:  files=`ls -1 *.txt`  for x in $files do     mv $x "`basename $files .txt`.text" done   I'm getting this error:  basename: too many arguments Try basename --help' for more information In terminal emulation applications, pressing CTRL + Left / Right arrows jumps from one word to the previous or next one. Is it possible to have the same functionality in a Linux console, whether it is in text or in framebuffer modes?  In my configuration, the CTRL + arrow keys are transformed into escaped character sequences and not interpreted. I've working LDAP authentication with the following setup   AuthName            "whatever"  AuthType            Basic  AuthBasicProvider   ldap  AuthLDAPUrl         "ldap://server/OU=SBSUsers,OU=Users,OU=MyBusiness,DC=company,DC=local?sAMAccountName?sub?(objectClass=*)"  Require ldap-group  CN=MySpecificGroup,OU=Security Groups,OU=MyBusiness,DC=company,DC=local   This works, however I've to put all users I want to authenticate into MySpecificGroup. But on LDAP server I've configured that MySpecificGroup also contains the group MyOtherGroup with another list of users.  But those users in MyOtherGroup are not authenticated, I've to manually add them all to MySpecificGroup and basically can't use the nested grouping. I'm using Windows SBS 2003.  Is there a way to configure Apache LDAP to do this? Or is there a problem with possible infinite recursion and thus not allowed? Escaping both forward slash and back slash with sed How to add features to a PostGIS layer using Quantum GIS? I am using Quantum GIS and the PostGIS dbms. I am able to call the PostGIS layer into QGIS but i am unable to add feature to it. I am using the method of adding features to a Shapefile but still it is not getting reflected in the layer, and I don't know how to save it to the PostGIS database if I am able to add the feature to the PostGIS layer.  I may not have used some technical term for this problem because i don't know what to call it, I am expressing the problem in my own terms. Let $X$ and $Y$ be independent $N(0; 1)$ random variables. Let $Z = \sqrt{X^2+Y^2}$.  (a) Derive the marginal pdf of $Z$ and then using the marginal pdf to compute ${\rm E}[Z^2]$  (b) Can you propose a different way other than that in (a) to compute ${\rm E}[Z^2]$  (c) Compute ${\rm E}[Z]$.  This is the whole question that I was asked. I can do (c) but I don't know how to find the marginal pdf of $Z$ with the given information and can't seem to find any formulas. Any help would be appreciated. I've been going through Shankar's Principles of Quantum Mechanics. In the section of the system of identical particles, he uses an example of billiards to illustrate the difference between identical particles in classical vs. quantum mechanics. He argues that in classical mechanics, we can track the history of a particle (a billiard ball) to distinguish it from another particle with no intrinsic differences. In quantum mechanics, however, he argues since continual observation is not possible, we can't use the same method to distinguish identical particles. A potential counter-example I thought was: suppose we have two non-interacting particles in the same square well. And at the end of some measurement, we find that one particle 1 is in a stationary state $\psi_1$, and particle 2 in $\psi_2$. We measure the system again after some time $t$, then we know whichever particle that's in $\psi_1$ must be particle 1 from the previous measurement. And the same goes for particle 2. Thus we can distinguish the two &quot;identical&quot; particles. What conceptual mistakes am I making here? Let $f,g:X\to Y$ be maps of spaces such that $f\simeq g$. Is it true that the mapping cones $\operatorname{cone}(f)$ and $\operatorname{cone}(g)$ are homotopy equivalent?  Can we write down an explicit homotopy equivalence? It ought not to be too hard to get a map $\operatorname{cone}(f)\to \operatorname{cone}(g)$ from the existence of a homotopy $H:X\times I\to Y$, but I haven't been able to find a sensible candidate, despite trying for quite a while. Suppose that $S_0$ and A are positive numbers, let $$S_{n+1} = \frac{1}{2}\left(S_n +\frac{A}{S_n}\right)$$ with $n \geq 0 $.      (a)Show that $S_{n+1} \geq \sqrt{A} $ if $n \geq 0$       (b)Show that $S_{n+1} \leq S_n $ , if $n \geq 1$      (c) Show that $s= \lim\limits_{n \rightarrow \infty} S_n$ exists      (d) find s   (a) Show that $S_{n+1} \geq \sqrt{A} $ if $n \geq 0$  Given $$P_n:  S_{n+1} = \frac{1}{2}\left(S_n +\frac{A}{S_n}\right) \geq \sqrt{A}$$  $$P_0: S_{1} = \frac{1}{2}\left(S_0 +\frac{A}{S_0}\right) \geq \sqrt{A}  $$  We assume that $P_n$ is true  $$P_{n+1}: S_{n+2}= \frac{1}{2}\left(S_{n+1} +\frac{A}{S_{n+1}}\right)$$ by assumption $$S_{n+2}= \frac{1}{2}\left(S_{n+1}\left(1 +\frac{A}{(S_{n+1})^2}\right)\right) \geq \frac{1}{2}\left(\sqrt{A}\left(1 +\frac{A}{(\sqrt{A})^2}\right)\right)$$ $$ S_{n+2}= \frac{1}{2}\left(S_{n+1} +\frac{A}{S_{n+1}}\right) \geq \sqrt{A}  $$  It follows that $S_{n+1} \geq \sqrt{A} $    (b) Show that $S_{n+1} \leq S_n $ , if $n \geq 1$  $$S_{n+1} \leq S_n$$ $$\frac{1}{2}\left(S_n +\frac{A}{S_n}\right) \leq S_n $$ Dividing by $S_n$ $$\frac{1}{2}\left(1 +\frac{A}{S_n^2}\right) \leq 1 $$ $$\frac{A}{2S_n^2} \leq \frac{1}{2}$$ $$A \leq S_n^2$$ $$S_n \geq \sqrt{A}$$ As  $S_{n+1} \leq S_n$ yields a true statement, it follows $S_{n+1} \leq S_n$ is true.    (c) Show that $s= \lim\limits_{n \rightarrow \infty} S_n$ exists  Since $S_{n+1} \leq S_n$, the sequence is non-increasing,   using the non-increasing theorem stating that      if $\{S_n\}$ is non-increasing then $$\lim\limits_{n \rightarrow &gt; \infty} S_n = \inf\{S_n\} $$     (d) find s    Is the argumentation in (a) and (b) appropriate? Also, I have to admit I m getting less confident in my argumentation (c) and (d). How to proceed in (c) and (d)?  Much appreciated for your input or help. MySQL: "UPDATE command denied to user ''@'localhost'" How do I share internet with Windows from my Linux box? I have a Linux (Ubuntu 12.04) PC connected to the internet with a Greenpacket WiMax USB modem. I want to share the Internet connection with another computer running Windows 7 Home Premium, connected to the Linux PC over a LAN. Is this possible? How? Is the reverse possible instead (connecting the internet to the Windows computer and sharing it with Linux)?