The math book i'm using states that the cross product for two vectors is defined over $R^3$:  $$u = (a,b,c)$$ $$v = (d,e,f)$$  is:  $$u \times v = \begin{vmatrix} \hat{i} &amp; \hat{j} &amp; \hat{k} \\ a &amp; b &amp; c \\ d &amp; e &amp; f \\ \end{vmatrix} $$  and the direction of the resultant is determined by curling fingers from vector v to u with thumb pointing in direction of the cross product of u x v.    Out of curiosity, what's the cross product if u and v are defined over $R^2$ instead of $R^3$ instead:  $$u = (a,b)$$ $$v = (d,e)$$  Is there a "degenerate" case for the cross product of $R^2$ instead $R^3$?  like this is some type of 2x2 determinant instead?  for instance if had a parameterization:  $$\Phi(u,\ v) = (\ f(u),\ \ g(v)\ )$$  and needed to calculate in $R^2$:  $$ D = \Bigg| \frac{\partial{\Phi}}{\partial{u}} \times \frac{\partial{\Phi}}{\partial{v}} \Bigg| $$  There are plenty of examples in the book for calculating the determinate D in $R^3$ but none at all for $R^2$ case.  As in:  $$ \iint_{V} f(x,y) dx\ dy = \iint_{Q} f(\Phi(u,v) \Bigg| \frac{\partial{\Phi}}{\partial{u}} \times \frac{\partial{\Phi}}{\partial{v}} \Bigg| $$  $$ \Phi(u,v)=(2u \cos v,\ \ u \sin v) $$
Wikipedia introduces the vector product for two vectors $\vec a$ and  $\vec b$ as $$ \vec a \times\vec b=(\| \vec a\| \|\vec b\|\sin\Theta)\vec n $$ It then mentions that $\vec n$ is the vector normal to the plane made by $\vec a$ and  $\vec b$, implying that $\vec a$ and  $\vec b$ are 3D vectors. Wikipedia mentions something about a 7D cross product, but I'm not going to pretend I understand that.  My idea, which remains unconfirmed with any source, is that a cross product can be thought of a vector which is orthogonal to all vectors which you are crossing.  If, and that's a big IF, this is right over all dimensions, we know that for a set of $n-1$ $n$-dimensional vectors, there exists a vector which is orthogonal to all of them.  The magnitude would have something to do with the area/volume/hypervolume/etc. made by the vectors we are crossing.  Am I right to guess that this multidimensional aspect of cross vectors exists or is that last part utter rubbish?
How to install Blender from the official website I installed Blender from official Ubuntu repository but its version is always behind the version available from the official Blender website.  I'm aware that I can launch the program within the downloaded folder, but i want to replace the install version with a new version downloaded from the website?   It must be present in the unity launcher.  I'm also aware of the available PPA's that are available, however I just want to install Blender manually.  Thanks in advance Now that all SE sites have their own meta site, why doesn't Stack Apps have a meta?  Sure, Stack Apps is somewhat of a limited site (or rather, a site about a limited subject), so by some counts, it might not need a meta. However, I think there are some things that could be discussed about how Stack Apps should be run and there is no good place to do that now. Thus, I support the creation of a Stack Apps meta. How closely can we estimate $\sum_{i=0}^n \sqrt{i}$ By looking at an integral and bounding the error? Why aren't people voting for questions? I have been texture painting in look dev mode. But when i try to render it, everything looks very dark. I have 3 sun lights in my scene, ambient turned on in Eevee settings. What am i missing? I really like the look of my scene in Look Dev mode, how can i achieve the same look when my scene is rendered too? Framing a question whose answer is an ordinal number I am the third daughter (or son) of my parents.   OR      How should a question that is answered with the above sentences be framed? What is the difference between ++i and i++? In C, what is the difference between using ++i and i++, and which should be used in the incrementation block of a for loop? Calculate the area of an irregular cyclic convex polygon Non-existence of $\lim \limits_{(x, y) \to (0,0)} \frac{x^3 + y^3}{x - y} $ How to show that $\lim \limits_{(x, y) \to (0,0)} f(x, y)$ does not   exist where,         $$f(x, y) =  \begin{cases} \dfrac{x^3 + y^3}{x - y} \; ; &amp; x \neq y \\  0 \; \;\;\;\;\;\;\;\;\;\;\; ; &amp; x = y  \end{cases}  $$   I tried bounding the value of the function as $(x, y)$ approaches $(0,0)$ but was not successful. I graphed the function and saw that it was actually approaching $0$ but Microsoft Math failed to render some points around the $x = y$ plane. So I imagine I need to find points $(x, y)$ such that $x$ is extremely close to $y$ but $(x, y)$ is not as close to $(0,0)$ to show that the function does not approach $0$ around the origin. But was unable to think of any.   Hints or suggestions would be awesome. Any help is appreciated. Why are integer literals with leading zeroes interpreted strangely? Can bug reports and feature requests requiring SE Dev attention be reported on per-site metas, or do they need to all go here? I want every new user from now on to have bash as their shell by default.    I know that to change your own shell to bash, you would use the command "chsh -s /bin/bash", but how do I automatically set all future users' shell to bash by default? How to remove \xa0 from string in Python? What's a way to implement a flexible buff/debuff system? Overview:  Lots of games with RPG-like statistics allow for character "buffs", ranging from simple "Deal 25% extra damage" to more complicated things like "Deal 15 damage back to attackers when hit."  The specifics of each type of buff aren't really relevant.  I'm looking for a (presumably object-oriented) way to handle arbitrary buffs.  Details:  In my particular case, I have multiple characters in a turn-based battle environment, so I envisioned buffs being tied to events like "OnTurnStart", "OnReceiveDamage", etc.  Perhaps each buff is a subclass of a main Buff abstract class, where only the relevant events are overloaded.  Then each character could have a vector of buffs currently applied.  Does this solution make sense?  I can certainly see dozens of event types being necessary, it feels like making a new subclass for each buff is overkill, and it doesn't seem to allow for any buff "interactions".  That is, if I wanted to implement a cap on damage boosts so that even if you had 10 different buffs which all give 25% extra damage, you only do 100% extra instead of 250% extra.  And there's more complicated situations that ideally I could control.  I'm sure everyone can come up with examples of how more sophisticated buffs can potentially interact with each other in a way that as a game developer I may not want.  As a relatively inexperienced C++ programmer (I generally have used C in embedded systems), I feel like my solution is simplistic and probably doesn't take full advantage of the object-oriented language.  Thoughts?  Has anyone here designed a fairly robust buff system before?  Edit: Regarding Answer(s):  I selected an answer primarily based on good detail and a solid answer to the question I asked, but reading the responses gave me some more insight.  Perhaps unsurprisingly, the different systems or tweaked systems seem to apply better to certain situations.  What system works best for my game will depend on the types, variance, and number of buffs I intend to be able to apply.  For a game like Diablo 3 (mentioned below), where nearly any bit of equipment can change a buff's strength, the buffs are just character stats system seems like a good idea whenever possible.  For the turn-based situation I'm in, the event-based approach may be more suitable.  In any case, I'm still hoping someone comes along with a fancy "OO" magic bullet which will allow for me to apply a +2 move distance per turn buff, a deal 50% of damage taken back to the attacker buff, and a automatically teleport to a nearby tile when attacked from 3 or more tiles away buff in a single system without turning a +5 strength buff into its own subclass.  I think the closest thing is the answer I marked, but the floor is still open.  Thanks to everyone for the input. I am trying to rotate a dual monitor which is attached to the VGA output of my laptop. My system is Ubuntu 15.10.  However, here is the outcome I get when I use xrandr:  xrandr --output VGA-1-0 --rotate right xrandr: output VGA-1-0 cannot use rotation "right" reflection "none"   I have seen other posts on this subject, which all recommend editing xorg.conf. I have tried that, too. Here's my current xorg.conf file:  Section "ServerLayout" Identifier "layout" Screen 0 "nvidia" Inactive "intel" EndSection  Section "Device" Identifier "intel" Driver "modesetting" BusID "PCI:0@0:2:0" Option "AccelMethod" "None" Option "RandRRotation" "True" EndSection  Section "Screen" Identifier "intel" Device "intel" EndSection  Section "Device" Identifier "nvidia" Driver "nvidia" BusID "PCI:1@0:0:0" Option "ConstrainCursor" "off" Option "RandRRotation" "True" EndSection  Section "Screen" Identifier "nvidia" Device "nvidia" Option "AllowEmptyInitialConfiguration" "on" Option "IgnoreDisplayDevices" "CRT" Option "RandRRotation" "True" EndSection   Still, I keep getting the same error when trying to use xrandr to rotate that screen.   Is there another way of doing so? Dealing with splits in my ground plane Whenever I'm making a cheap, 2-sided PCB, I run into this problem:    Where I have signal lines (on the top layer here) running over top of a big fat power trace (on the bottom layer).  Now, this is obviously terrible because there is no good return path on the bottom side.  For issues like this, Hott consultants has a good recommendation:    However, I wonder if a solution like the one below is also acceptable.    My intuition tells me yes, this solution is ok.  Over the break in the ground plane, the current will hop over the vias and then back down to the lower plane.  Some back of the envelope math tells me that this adds a trivial amount to the loop area, and therefore a trivial amount to the return impedance.  Can someone tell me...   How right / wrong my intuition is here about using vias? If my intuition is wrong, why? Also, if my intuition is wrong, what is the correct solution in this sort of situation (2 layer, with power traces under signal)?   For the record, these particular signal lines aren't terribly fast (500 kHz), but I would like to know how to deal with this situation when I have signals in the 5 - 10 MHz range.  Edit  Below is a screenshot of about 60% of the board.  Maybe it will help with the critique How do I download torrents? List of all "partitions"/ "cuttings" of a list Biblatex authoryear dashed for entries with no year I'm using biblatex authoryear and have multiple entries of the same author that do not have a year. When I cite them there are letters shown to distinguish them:   But in the bibliography those letters are not shown and you can only distinguish them by the title. Is it possible to show the letters in the bibliography, too?  \documentclass[a4paper,12pt]{article}% \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{filecontents} \begin{filecontents}{bib.bib} @BOOK{bar1,   title = {Text One},   year = {},   author = {Firstname Lastname}, } @BOOK{bar2,   title = {Text Two},   year = {},   author = {Firstname Lastname}, } \end{filecontents}  \usepackage[backend=biber, style=authoryear-icomp, dashed=true]{biblatex} \addbibresource{bib.bib}  \begin{document}  \cite{bar1} and \cite{bar2}  \printbibliography \end{document} Convergence of $\sum_{n=1}^\infty\frac{n}{(n+1)!}$ Can someone give an explanation using the definition of convergence in partial sum to show how the above infinite sum converges to 1?  Thanks