Component of Component of a vector
Why taking components of a component of a vector is invalid?
Why is the PHP version different in phpinfo() and CLI? I realized that there is a miss matching between the PHP version reported by phpinfo() and php -v (in the CLI).  phpinfo():  5.5.24 php -v: 5.6.9   I am working on a Mac OS X 10.10 (Yosemite) and installed a library (php-version) to try to manage the PHP versions.  It reports me 3 different versions installed on my laptop:    5.4.41   5.5.25 * 5.6.9   I would like to set the same version of PHP in both Apache and CLI.  How can I tell Apache which PHP version use?  I tried to write this line in my httpd.conf:  LoadModule php5_module    /usr/local/opt/php56/libexec/apache2/libphp5.so   Then:  sudo apachectl restart   But all this not solve the problem… In my PHP info I still have: 5.5.24 Solution of trigonometric equation $0 = x\cos(x)+2$ In an HTML table, the cellpadding and cellspacing can be set like this:  &lt;table cellspacing="1" cellpadding="1"&gt;   How can the same be accomplished using CSS? Incorrect whitespaces or list item size in Android SE app Noticed that sometimes when there are 10+ answers to a question - some answers look like they have \n\n...\n line breaks in the end, so the body of the answer has incorrect height.  It probably has something to do with ViewHolder or adapter not setting correct row height. Although as far as I know, it should do this according to text amount/length. Arduino Negative Voltage Reading with Positive Ground I am trying to measure voltage &amp; current off a solar charge controller from panel, battery &amp; load simultaneously with my arduino. I have managed to achieve this by tying all the negatives together and using it as GND, however, when I do that the charge controller doesn't function properly and when I switch to off-mode the loads still run. After checking the instruction manual further, I found out that the positives are shared common between Pan, Bat, Load. This was confirmed with a multimeter.   So now what I am thinking of doing is using the positive from the battery as the GND reference and then measuring all other voltages (through voltage dividers with respect to the GND). The voltages should then range between -17V to 0V.   However, I have read that the analog input pins are not able to read negative voltages, and with respect to the positive GND, the Analog inputs will all be negative.   EDIT: I was thinking of using an amplifier to invert positive ground to a negative ground, meaning I could measure all the positive sides of the voltages. But after getting answers from you guys it seems I may not have to use an amplifier and a voltage divider may suffice. I have currently included a 2 resistor voltage divider between each analog input, +5V source, Vmeasure which you can see in the diagram below.  According to my calculations, if this works, @-17V I should see 0.6V @ analog input. @0V I should see 4V at analog input.   I'd appreciate feedback on this circuit, and below in the second image I've also included a piece of another method suggested by @jonk (thanks), and would be interested in knowing what the benefits are of method B) over method A)  Thanks!  Ps. the ameters are there in place of ACS712 current sensors which also go into analog input pins.   Method A)        &ndash; Schematic created using   Method B (EDIT: This method has worked, I am still yet to test method A) thanks @Jonk)) Ubuntu 17.10 Keyboard not working after login following upgrade from 17.04 (Gnome version) How to make a half cylinder? I am having a tough time building a half cylinder and have not found any tutorial showing me how to do that. Concatenate multiple files with two blank lines as delimiter? I'm trying to join Markdown files with two newlines between each file.   I've tried using the following command:  paste -sd '\n\n' file1.md file2.md   This joins the files with a newline, but not two newlines, as I need.   What command can I use to join an arbitrary number of files with two \n characters between each join? Can you use Uncanny Dodge against extra damage effects? Is it possible to be admitted to a masters in Germany with a 3-year bachelors from another country? What content type to force download of text response? I need to stream a file to the Response for saving on the end user's machine.  The file is plain text, so what content type can I use to prevent the text being displayed in the browser? When should I use GET or POST method? What's the difference between them? From this  by orlp:     If I strap a rocket booster to a rocket in space and fire it for one second, then the work provided is much higher when the rocket is flying fast compared to when the rocket was stationary. In both cases the rocket fires for the same duration but in the former case the rocket travels a much greater distance during this period. What gives? I was going through fluid dynamics and I came upon this section on the shape of a liquid surface in a rotating cylindrical vessel. The working in the book is something like this:   It considers a mass $m$ of water on the surface of the rotating liquid at a distance $x$ from its axis of rotation .It takes into account a pseudo force ($m \cdot x \cdot \omega^2$) radially outwards and its weight force ($mg$).It then proceeds to calculate the net force and the angle it makes with the vertical.From there through basic integration it gets the equation of the surface of liquid  i.e. $y= \dfrac{(x^2 \cdot \omega^2)}{2g}$ .  I get this working.  But if I were to view the same rotation through an inertial frame,how would I proceed? I mean, there wouldn't be any pseudo force, rather a centripetal acceleration of same magnitude radially inwards.This is where I am facing certain doubts.  The forces on the liquid would be  the centripetal force inwards and the weight force downwards.The net force would therefore be in a different direction,and consequently the slope of the surface at that point would be in a different direction.How is it possible for the same surface to have two different shapes in two different frames?   I would be obliged if someone could help me out by pointing out the flaw in my understanding. Evaluating $\lim_{n \to \infty} (1 + 1/n)^{n}$ I was recently thinking about how I could evaluate the famous limit of 'e' as I haven't ever seen a proof. I can't really find anything online so I've tried to evaluate the limit myself. And I was also thinking it would be nonsensical to use L'Hopital's rule, am I right?  So I did the following:  $$\lim_{n \to \infty} \left(1 + \frac{1}{n} \right)^{n} = \exp \left(\lim_{n \to \infty} n \cdot \ln \left(1 + \frac{1}{n} \right) \right)$$  $$=\lim_{n \to \infty} \exp \left( n \cdot \left( \frac{1}{n} - \frac{(1/n)^{2}}{2} + \cdots \right) \right)$$  $$= \lim_{n \to \infty} \exp \left( 1 - \frac{(1/n)}{2} + \cdots \right)$$  $$= e$$  I am not sure, is my logic correct or does it create circularity by taking logarithms and assuming $$f(x) = \exp \left( \ln f(x) \right)$$ ? New breaker and panel: What's the brown goo where they connect? I tried this way, I only need to know if this is correct or if there are better ways to solve this:  $2^{1000}$ does not have a factor of $5$ obviously therefore we can assume  $$ 10^{m} &lt; 2^{1000} &lt; 10^{m+1}$$ for some  $m$   Assume $ k = 2^{1000}$, then take log on both sides $\log k = 1000  \log 2 \approx 301.02999 &gt; 301$  Therefore $2^{1000}$ has $302$ digits. How to show that $\sum\limits_{k=0}^n (-1)^k\tfrac{{ {n}\choose{k}}}{{ {x+k}\choose{k}}} = \frac{x}{x+n}$ I'm trying to show that      $$\sum_{k=0}^n (-1)^k\frac{{ {n}\choose{k}}}{{ {x+k}\choose{k}}} = \frac{x}{x+n}$$   I've tried expanding this using definition of binomial coefficient, then binomial expansion, now I'm using induction to simplify $\sum_{k=0}^m (-1)^k\frac{{ {n}\choose{k}}}{{ {x+k}\choose{k}}}$, but it looks very inelegant, is there a better way? Why are random walks intercorrelated? Examples and further results about the order of the product of two elements in a group