A vector quantity has both magnitude and direction. So, for any physical quantity to be a vector, it should have a direction and a magnitude. Though this is a necessary condition for any quantity to be a vector it is not sufficient. To qualify as a vector a quantity must also follow the laws of vector addition.  Now getting to the point, the most commonly quoted reason to explain why force is a vector is that it has direction and magnitude. Most textbooks say nothing more than this. How can one mathematically prove that force is a vector?
In most of the mechanics textbooks , the  is stated without any justification or an experiment to confirm this law. Also after introducing this law forces are also decomposed without any justification. Please suggest any links or books related to this. Any help will be appreciated. Thanks in advance.
How to install Flash player on Ubuntu? Intuition on proof of Cauchy Schwarz inequality To prove Cauchy Schwarz inequality for two vectors $x$ and $y$ we take the inner product of $w$ and $w$ where $w=y-kx$ where $k=\frac{(x,y)}{|x|^2}$ ($(x,y)$ is the inner product of $x$ and $y$) and use the fact that $(w,w) \ge0$ . I want to know the intuition behind this selection. I know that if we assume this we will be able to prove the theorem, but the intuition is not clear to me. Inspired by a StackOverflow question ().  Is there a DBMS that actually supports the = NULL syntax? In the film Thor, Odin casts a spell on Mjolnir and strips Thor of his power.  He then banishes Thor to Midgard (Earth).  On Earth, Thor is seemingly just a strong and durable human.  Later in the film, he becomes worthy, and regains control of Mjolnir.  This then grants him his full power set and control over Mjolnir.  My question is, prior to this, could Thor control lightning all on his own?  Or is this purely an ability of Mjolnir? How do I efficiently filter computed values within a Python list comprehension? The Python list comprehension syntax makes it easy to filter values within a comprehension.  For example:  result = [x**2 for x in mylist if type(x) is int]   Will return a list of the squares of integers in mylist.  However, what if the test involves some (costly) computation and you want to filter on the result?  One option is:  result = [expensive(x) for x in mylist if expensive(x)]   This will result in a list of non-"false" expensive(x) values, however expensive() is called twice for each x.  Is there a comprehension syntax that allows you to do this test while only calling expensive once per x? How do I tokenize a string in C++? Java has a convenient split method:  String str = "The quick brown fox"; String[] results = str.split(" ");   Is there an easy way to do this in C++? pdf of a quotient of uniform random variables Suppose $x_1, x_2$ are IDD random variables uniformly distributed on the interval $(0,1)$.  What is the pdf of the quotient $x_2 / x_1$? Identify $P(2014)$ if $P(i)=1/i$ for every positive integer $1\le i\le2013$ Suppose $P(x)$ is a polynomial of degree $2012$ and $P(x) = 1/x$ when $x$ takes the integer values $1\cdots2013$ (inclusive). What is the value of $P(2014)$?   I get $1/1007$ but I'm not sure if it's right, and my method is very inelegant. Can we omit a duplicated preposition? Is on needed a second time in this sentence?        There is a bad effect either on human health or environment.    There is a bad effect either on human health or on environment. Left ideals of $M_n(K)$ Shadow Incorrect at Bottom of Edge In the images below, I have a cube that is sticking up about two thirds through a plane. But, despite the fact that the cube is not hovering over the plane, the shadow seems to indicate otherwise. How can I fix this shadow? The "New Contributor" dialog is masked The New Contributor dialog is masked from where the comment section ends. Here are some screenshots just to be more clear: With no comments:    With one comment:    No masking with more comments: How to stop an animation in C# / WPF? I have something like this:   barProgress.BeginAnimation(RangeBase.ValueProperty, new DoubleAnimation(     barProgress.Value, dNextProgressValue,     new Duration(TimeSpan.FromSeconds(dDuration)));   Now, how would you stop that animation (the DoubleAnimation)? The reason I want to do this, is because I would like to start new animations (this seems to work, but it's hard to tell) and eventually stop the last animation... Given any $10$ consecutive positive  integers , does there exist one integer which is relatively prime to the product of the rest ? I know how to compute modular multiplicative inverses for co-prime variables $a$ and $b$, but is there an efficient method for computing variable $x$ where $x &lt; b$ and $a$ and $b$ are not co-prime, given variables $a$, $b$ and $c$, as described by the equation below?   $ a x \equiv c \mod b $   For example, given  $ 154x \equiv 14 \mod 182 $, is there an efficient method for computing all the possibilities of $x$, without pure bruteforce?  Please note that I'm not necessarily asking for a direct solution, just a more optimized one.    I do not believe that the Extended Euclidean Algorithm will work here, because $a$ and $b$ are not co-prime.   Edit:  Follow up question, since the first one had a shortcut:  Could the be computed efficiently as well?  $12260x \equiv 24560 \mod 24755$.  $107$ needs to be one of the computed answers. The sign in the formula for the voltage across the inductor This question is about voltage induced in a inductance.  I know the rate of change of current induces EMF in the inductor. The direction of the induced EMF will be opposite to the change that is causing it. The EMF induced is given by the following formula.  $$V_l = -L \frac{dI}{dt}$$  The negative sign indicates direction of induced emf across the coil is opposite the current change.     In the picture I have drawn current direction and \$V_l\$ direction.  If \$\dfrac{dI}{dt}\$ is postive, \$V_l\$ will be negative according to formula right? Or does the direction of \$V_l\$ has to be reversed?  In many Internet references I found \$V_l = L \dfrac{dI}{dt}\$. Which one is correct?  I hope they reversed \$V_l\$ reference points and removed the negative sign in the formula. Please clarify. "SELECT *" why is it an antipattern On multiple questions here and on stack overflow i saw people saying in the comments and answers that select * from table is almost always an antipattern without any explaination why. Althrough I can sort of deduce why it is an antipattern. I may be looking over a detail that someone else with better understanding of the issue noticed.  So here is my question, why people say that select * is an antipattern. I'm new to TDD and xUnit so I want to test my method that looks something like:   List&lt;T&gt; DeleteElements&lt;T&gt;(this List&lt;T&gt; a, List&lt;T&gt; b);   Is there any Assert method that I can use ? I think something like this would be nice  List&lt;int&gt; values = new List&lt;int&gt;() { 1, 2, 3 }; List&lt;int&gt; expected = new List&lt;int&gt;() { 1 }; List&lt;int&gt; actual = values.DeleteElements(new List&lt;int&gt;() { 2, 3 });  Assert.Exact(expected, actual);   Is there something like this ? I am needing to auto increment a field based on groups within a feature class. I have 8 plots within a given polygon and I need to assign them an ID from 1-8 for each set of plots within each polygon. The polygon would have its own unique ID number to be used to group the plots.  I assume it would be an alteration of this:  rec=0 def autoIncrement():  global rec  pStart = 1   pInterval = 1   if (rec == 0):    rec = pStart   else:    rec = rec + pInterval   return rec This has always been a big want for me. I love using multi-monitor setups but only want a single laptop. Does the 2011 MBP Thunderbolt support chaining multiple monitors together? I know that you can add a monitor and say, a few hard drives, but can you connect it to 2 27" monitors and your flat screen TV (for example)? Thanks!