I have not worked with 3d coordinate systems for a while and this is giving me a bit of a struggle. I have the coordinates of four points in XYZ. Two points I use to generate the line AB and the other two I use to generate line CD. I would like to find the closest XYZ coordinate on AB to line CD and the closest XYZ coordinate on CD to line AB. My idea was to find the equation for the perpendicular line between AB and CD and then find the intersection of this new line with AB and CD. However, I do not really know where to begin. If anyone could help that would be very much appreciated
I have two arbitrary lines in 3D space, and I want to find the distance between them, as well as the two points on these lines that are closest to each other.  Naturally, this only concerns the skew case, since the parallel and intersecting cases are trivial.  I know how to find the distance, as the question was asked before and answered .  I haven't found a good explanation on how to find the two points that determine that distance, though.  So specifically, given two lines  $$L_1=P_1+t_1V_1$$ $$L_2=P_2+t_2V_2$$  I would like to find two points $X_1$ on $L_1$ and $X_2$ on $L_2$ such that the distance between $X_1$ and $X_2$ is minimal.
How can I restart a Windows 7 computer without screen or mouse? How to show that  $$ \lim _{\alpha \rightarrow \infty } \sup_{t \in \left [0,T \right]} \left | e^{-\alpha t} \int _ 0 ^t e^{\alpha s} ~ dB_s \right | =0, \ \ \text{a.e.} $$  where $\left (B_s \right)_{s\geq 0}$ is a real standard brownian motion starting from zero ? All the info about TLP on the surface looks more like advertisement and marketing to me, I can't see any detailed comparison to default configurations. What are the precise differences compared to default (K/L)Ubuntu installs under the hood, are they really effective?  Update: And if they really are, why aren't they a part of a default config? What is the difference between a field and a property? Proving that an integral domain has at most two elements that satisfy the equation $x^2 = 1$. Is there an easy way to convert a number to a word in PHP? How to split a very long equation (using AMSMATH package)? Are there instances where citing Wikipedia is allowed? For example, suppose you're writing an article about triangles, and you want to include a short proof of Pythagoras' theorem, but you can't quite remember how to derive it, so you pop over to:  and check the proof there. It works mathematically, makes sense and is as far as you can tell, correct, so you include it. Now the proof on Wikipedia doesn't have a citation because probably someone has been taught it and copied it over to Wikipedia, but it's still correct. The only thing you could cite for the proof would be Wikipedia, would a citation be ok or should it just be copied down without a citation?  Edit: Ok, there have been many responses. Most with a firm "no", but some giving the reason: "Just get the proof from a book" since it's so common, but suppose then it wasn't a common proof of something. Maybe it's not very well known or something but suppose there is a small Wikipedia page with the correct mathematical proof. How much effort are people going to go to to find a proof in a book when there's a (correct (which we know, becuase it's maths and we can check it)) proof on Wikipedia? Why is it so bad to say: "I found this proof on Wikipedia, and the maths checks out so it doesn't matter who wrote it, it is correct, but that's where I found it." Or if you do find it somewhere else why not say: "Proof taken from Wikipedia and verified by the proof shown in "Triangles and their properties, Nature, 2014, p113 etc..." Derivative of conjugate multivariate function Is there a place to dispute duplicate markings? I recently upgraded to windows 10 on my daily driver laptop and I have liked it so far except for one thing. Randomly the windows menu decides to stop working when I press the windows key or I press the button in the corner. Has anybody else experienced this problem? Solutions?  Thanks! I read news on the BBC (on its website) this morning. When I was reading the news I noticed the definite article the was missing in a sentence. As per my opinion, the reporter missed the article the. But I also thought that the reporter also had some point as he did not use the article there. So I just wanted to confirm so landed here to get some help on it.     On the website:      It also proclaimed the group's leader, XYZ  (name), as caliph and   "leader for Muslims everywhere".      As per my opinion it should have been:      It also proclaimed the group's leader, XYZ  (name), as the caliph and   "leader for Muslims everywhere".   Source link: Converting a DC motor to a generator Can ANY DC motor become a generator if torque is applied to the shaft? example:   And by any, I mean the really cheap ones too.  Or is there a specific type of motor (of which I do not know the name) that can work in both directions (battery to power the shaft OR torque-on-shaft to create a voltage difference between the terminals). $ \lim_{n \to \infty} \left(\frac 1{n^2+1}+\frac 2{n^2+2}+\frac 3{n^2+3}+\cdots +\frac n{n^2+n}\right)$ Evaluating string "3*(4+2)" yield int 18 Is there a function the .NET framework that can evaluate a numeric expression contained in a string and return the result? F.e.:  string mystring = "3*(2+4)"; int result = EvaluateExpression(mystring); Console.Writeln(result); // Outputs 18   Is there a standard framework function that you can replace my EvaluateExpression method with? In shell scripts used for unit testing with dynamic libraries in a directory other than the typical @rpath, I have previously been able to set DYLD_FALLBACK_LIBRARY_PATH to set the directory containing the libraries. Under 10.11.1, bash seems to ignore attempts to set this environment variable:  $ sh -x testscript.sh + DYLD_FALLBACK_LIBRARY_PATH=/Users/something/testinglibs + export DYLD_FALLBACK_LIBRARY_PATH + exec printenv   and DYLD_FALLBACK_LIBRARY_PATH is not present in printenv's output.  Is this a security-related hack in 10.11's shell? I haven't been able to find this change documented in man pages or online. Uniform continuity of $f(x) = x \sin{\frac{1}{x}}$ for $x \neq 0$ and $f(0) = 0.$ For the $f(x) = x \sin{\frac{1}{x}}$ for $x \neq 0$ and $f(0) = 0,$ my text book asks the following questions.  (b) Why is $f$ uniformly continuous on any bounded subset of $\mathbb{R}$?   (c) Is $f$ uniformly continuous on $\mathbb{R}$??  The graph for the function is this.    For the question (b), if I take subset between $[0.2,0.6]$ or the subset where the slope is steep, I don't think the function is uniformly continuous because I think for a given $\epsilon&gt;0$, there is no unique $\delta &gt;0$ for the bounded subset. Therefore, it also cannot be uniformly continuous on $\mathbb{R}.$ However, the questions sounds like the function is uniformly continuous and the book says that it is uniformly continuous. The answer on the book says something but I need more explanation. Thanks. In  (the original movie), Marty comes back from 1955 10 mins early to prevent Doc from being shot (which he fails at). He then rolls up to the Lone Pine Mall and sees himself hit 88 mph and shoot back to 1955 (again). Marty then carries on with this life as usual. What happens to the Marty that goes back to 1955 at the end of the film?  This obviously is a recursive problem where it would create another timeline and Marty would keep coming back 10 mins early to view another Marty zip back to 1955? I dont quite understand how pipes work in bash.  I know that it takes an output from one command as the input in another command.  What an output is I can get because it's what the command prints out to the screen.  But how do I know what input a command will take?  Here is an example I thought would work:  which gem | rm   Unfortunately it didn't.  Which gem prints out /usr/bin/gem so that must be the output right?  I thought that was given to rm so it woud be rm /usr/bin/gem but I was wrong.  So my question is, how do I know what input a command takes? Detecting Color of iPhone/iPad/iPod touch?