A triangle has corners at $(0,0), (3,4)$, and $(2,-1)$. Compute its area using a determinant.  I'm not sure how to set up the matrix to use a determinant to find the area. Thanks for any help!!
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   $$M = \begin{pmatrix} 1 &amp; 1  &amp; 1\\  x_1 &amp; x_2 &amp; x_3\\  y_1 &amp; y_2 &amp; y_3 \end{pmatrix}$$
Can the program counter on Intel CPUs can be read directly (that is without 'tricks') in kernel mode or some other mode? No such file or directory? But the file exists! I've downloaded a game (Shank) but the bin file doesn't run. The error that is shown when I try to launch the executable is:  bash: ./shank-linux-120720110-1-bin: No such file or directory Which graphics cards are supported by the new AMDGPU driver in Ubuntu 16.04? The new open source AMDGPU driver pre-installed in Ubuntu 16.04 should support all the "Volcanic Islands" cards out-of-the-box. This means that AMD Radeon R9 285, R9 380/380X and R9 Fury/FuryX graphics cards will be supported out-of-the-box if Ubuntu 16.04 LTS is installed.  However I read that the open source AMDGPU driver has an experimental support for the "Sea Islands" graphics cards (HD7790, R7 260, R9 290, R7 360, R9 390). Is this support enabled by default on Ubuntu 16.04, or will these graphics cards use the Radeon driver?  Thanks Disable App Update Notifications on OSX? Formal basis for variable substitution in limits So we do a lot of variable substitution in limits at school. Stuff like $\lim\limits_{x\to5}\ (x+1)^2\ =\ \lim\limits_{y\to6}\ y^2$, where we define the substitution $y = x + 1$.  But I've never been clear on what exactly the theoretical basis for this is. What is the formula that you're actually applying when you do variable substitution? What are the formal conditions under which it is possible?  My conjecture would be the following:  For all continuous $f$, and all real $a$: $\lim\limits_{x\to a}\ (f\circ g)(x)\ =\lim\limits_{x\to g(a)}\ f(x)$, where $g$ is a continuous function  So to take my first example, $f$ would be $x^2$, $g$ would be $x + 1$, and $a$ would be $5$.  Am I in the right area? If this is correct, can it be proven using $\epsilon$-$\delta$? I had a half-hearted shot at it the other night and didn't get anywhere. How can I make a procedural polished granite material? package makeidx: print first letter for each group of keys I've created index with makeidx package. It works. What really need is to print the first letter above every group of keys (i.e. referring to the image I want to print the red letter, that I have manually inserted). Completely uninstall an app on Lollipop The replier's From email address is modified when Marketing Cloud RMM forwards an email to the routing address We've set up RMM to forward replies to routing address and SFMC adds our subdomain to the replier's From email address (i.e. replier@gmail.com becomes replier.gmail.com@oursubdomain.com). However, their reply-to address remains the same.  It seems like DMARC Forward Reply Rewrite is enabled, but we were advised not to touch it as that may affect email deliverability.  Is there any workaround to this? And what will happen if we disable this feature? Enabling GPU rendering for Cycles? I want to enable GPU rendering, but there is no option in User Preferences > System:    Why is this? How can I get cycles to render using my GPU? Number of permutations of $n$ elements where no number $i$ is in position $i$ Consider the following snippet:  $$\text{d}_{H}(A,B)  = \max\left\{ \sup_{a\in A} \inf_{b\in B} \text{d}(a,b),\sup_{b\in B} \inf_{a\in A}\text{d}(a,b)\right\}$$     In the output, the text under the inf's is significantly smaller then the one under the sup's. The same happens when I have it in an align environment.  You can see the same problem in the wiki definition of the . How can I have the typeset under the inf's and sup's in the same size? $[S]$ means linear span of S. Then $[S] = [[S]]$ My definition of span:   Suppose a vector space $(V,+,\cdot)$, and       $$S = \{u_1,\cdots,u_n\}$$      (and $S$ is a subset of $V$, not a subspace)      $$[S]=:\bigcap_{W\subset V, W\supseteq S} W$$      In other words, $[S]$ is, by definition, the intersection of all $W$,   such that $W$ is a subspace of $V$ and $W$ contains $S$.   Well, then I need to prove that   $$[S] = [[S]]$$  My strategy:  $[[S]]$ means the intersection of all subspaces that contain $[S]$. If all the subspaces contains $[S]$, so does their intersection. Therefore, $[S]\subset[[S]]$  However, I'm having trouble proving the converse:  $$[[S]]\subset[S]$$  Any hints? What exactly is the point of memoryview in Python I've recently started playing Minecraft, and I only have a little experience with enemies (I've been playing with the peaceful setting so I can figure out what the heck I'm doing).  When I did play some survival mode, I happened to be at the top of a pretty large cliff when night hit, and this seemed to protect me - the baddies just piled up at the bottom of the cliff.  However, it seems like this probably only worked due to pathing flukes.  The terrain behind me was pretty easy going, so I suspect if enemies spawned in the right areas, they could still come up and visit.  That said, how can I set up a quick protective shelter for the first night or two?  The days don't seem to last very long, so I don't expect I would be able to generate a lot of high-end materials, tools, etc. to work with.  Also, the zombies caught fire nicely when the sun came up, but there were still some spiders and some other things (creepers? - they were explosive) that didn't go away at daybreak.  Do I need multiple exits for my shelter or some other way to avoid running into all the enemies outside the front door that have accumulated overnight? In a closed system, I know the rider wouldn't be able to change the combined center of gravity of the system by moving his weight around. If he pushes the bike to lean to the right, he would go to the left by the amount that would keep the combined cog the same.  But a bike with tires and a road under it is not a closed system, the tires wouldn't let the lower part of the bike left or right when tryingo to initiate lean. Does this mean it is possible to push with your bodies inertia to the bike, and the combined cog will be changed? I thing if you do so above the cog of the bike, you could, but I am not sure.  Edit: My question is not about if you can move the center of gravity of the bike at all. My question is about if can you change the combine cog of the system by moving around on the bike?  English is not my mother tongue, sorry for the confusion Evaluate  $$\lim_{n\to\infty} \left(\frac{1^p+2^p+3^p + \cdots + n^p}{n^p} - \frac{n}{p+1}\right)$$ Let $X$ be a smooth projective variety over $\mathbb{C}$, then the Neron-Severi group $NS(X)$ of $X$ is defined to be the Picard group of $X$ modulo algebraically equivalent relations.   On the other hand, by the exponential sequence, there is a first Chern class map  $$c_1: {\rm Pic}(X) \to H^2(X, \mathbb{Z}).$$ It is claimed that the image of $c_1$ coincides with $NS(X)$.   I want to know why this is true. Any suggestion or reference is greatly welcome! How do I backup my Ubuntu Phone? What does the explicit keyword mean?