What is the $\hat{r}$ (vector) in the formula $\vec{E} = k\frac{q}{r^2} \hat{r}$ for the electric field ? Why we dont use the vectors $\vec{i},\ \vec{j},\  \vec{k}$? Also why this vector doesn't appear in the formula where field is the gradient of potential where we have $\vec{i},\ \vec{j},\  \vec{k}$ (unit vectors)?
How can I find the unit vector of a three dimensional vector? For example, I have a problem that I am working on that tells me that I have a vector $\hat{r}$ that is a unit vector, and I am told to prove this fact:     $\hat{r} = \frac{2}{3}\hat{i} - \frac{1}{3}\hat{j} - \frac{2}{3}\hat{k}$   I know that with a two-dimension unit vector that you can split it up into components, treat it as a right-triangle, and find the hypotenuse. Following that idea, I tried something like this, where I found the magnitude of the vectors $\hat{i}$ and $\hat{j}$, then using that vector, found the magnitude between ${\hat{v}}_{ij}$ and $\hat{k}$:     $\left|\hat{r}\right| = \sqrt{\sqrt{{\left(\frac{2}{3}\right)}^{2} + {\left(\frac{-1}{3}\right)}^{2}} + {\left(\frac{-2}{3}\right)}^{2}}$   However, this does not prove that I was working with a unit vector, as the answer did not evaluate to one. How can I find the unit vector of a three-dimensional vector?  Thank you for your time.
Going to Pakistan in 3 weeks time but my NICOP is expired and my son's hasn't arrived yet My NICOP is expired and I have already booked a flight to Pakistan via Qatar Airways. My wife has a valid NICOP card but my 5 years old child  has never had a NICOP or any other Pakistani ID card other than the British passport. I have made an appointment for next Monday to renew my NICOP and apply for my child's new NICOP card. My question is, what if the cards of my child and mine don't arrive in time, can we still travel to Pakistan, given that we have the receipts as a proof of application? I made a Yu-Gi-Oh card but it is not broken like current broken Yu-Gi-Oh cards and here it is to see but please before you make your decision about it take another look on ''dragun of red eyes''. How prove this inequality $\frac{a}{b+3}+\frac{b}{c+3}+\frac{c}{d+3}+\frac{d}{a+3}\le 1$ Question:  let $a,b,c,d\ge 0$,such $$a^2+b^2+c^2+d^2=4$$ show that $$\dfrac{a}{b+3}+\dfrac{b}{c+3}+\dfrac{c}{d+3}+\dfrac{d}{a+3}\le 1$$  My try: By Cauchy-Schwarz inequality,we have $$\sum_{cyc}\dfrac{a}{b+3}\le\sqrt{(\sum_{cyc}a^2)\left(\sum_{cyc}\dfrac{1}{(b+3)^2}\right)}$$ then we have only prove this $$\sum_{cyc}\dfrac{1}{(a+3)^2}\le \dfrac{1}{4}?$$ This is  not true,in fact, we have $$\sum_{cyc}\dfrac{1}{(a+3)^2}\ge \dfrac{1}{4}?$$ because we have $$\dfrac{1}{(a+3)^2}\ge\dfrac{5-a^2}{64}$$ this is true because $$\Longleftrightarrow \dfrac{(a-1)^2(a^2+8a+19)}{64(a+3)^2}\ge 0$$ so $$\sum_{cyc}\dfrac{1}{(a+3)^2}\ge\sum_{cyc}\dfrac{5-a^2}{64}=\dfrac{1}{4}$$ can see:  This methods is from: can  see:  By the way I have see this three variable inequality  let $a,b,c$ be non-negative numbers such that $$a^2+b^2+c^2=3$$  show that $$\dfrac{a}{b+2}+\dfrac{b}{c+2}+\dfrac{c}{a+2}\le 1$$  proof: By expanding,the inequality becomes $$ab^2+bc^2+ca^2\le abc+2$$ without loss of generality,assume that  $$\min(a,b,c)\le b\max(a,b,c)$$ then \begin{align*} 2-ab^2-bc^2-ca^2+abc&amp;=2-ab^2-b(3-a^2-b^2)-ca^2+abc\\ &amp;=(b^3-3b+2)-a(b^2-ab+ca-bc)\\ &amp;=(b-1)^2(b+2)-a(b-a)(b-c)\ge 0 \end{align*} Equality occurs for $(a,b,c)=(1,1,1)$ and also for $(a,b,c)=(0,1,\sqrt{2})$ or any cyclic permutation.  so my Four-inequality  variable inequality,can use this methods $$\dfrac{a}{b+3}+\dfrac{b}{c+3}+\dfrac{c}{d+3}+\dfrac{d}{a+3}\le 1$$ $$\Longleftrightarrow a^2cd+3a^2c+3a^2d+9a^2+ab^2d+3ab^2+abc^2-abcd+3ac^2+9ac+3b^2d+9b^2+3bc^2+bcd^2+3bd^2+9bd+9c^2+3cd^2+9d^2-81\le 0$$ $$\Longleftrightarrow a^2cd+3a^2c+3a^2d+ab^2d+3ab^2+abc^2+3ac^2+9ac+3b^2d+3bc^2+bcd^2+3bd^2+9bd+3cd^2\le 45$$ $$\Longleftrightarrow a^2(cd+3c+3d)+ab^2d+3ab^2+abc^2+3ac^2+9ac+3b^2d+3bc^2+bcd^2+3bd^2+9bd+3cd^2\le 45$$ $$\Longleftrightarrow (4-b^2-c^2-d^2)(cd+3c+3d)+ab^2d+3ab^2+abc^2+3ac^2+9ac+3b^2d+3bc^2+bcd^2+3bd^2+9bd+3cd^2\le 45$$  Then  I can't  Thank you  very  much When to use ' (or quote) in Lisp? Purpose of hole in stove leading to oven I'm staying in a hotel and have access to a "electric coil stove" for one of the first times.  I accidentally turned on the oven light and saw that the light was shining up through only one of the stoves:    There's only a port from the oven into the top right stove, and note that the stove is not on.  Here's a view from within the oven:    Is there a hole here as a vent for the oven for some reason, or is this for the stove? I typically use a glass "flat-top stove" that doesn't have an oven underneath it. I've never seen a direct hole from a stove to the oven before, and have to wonder what the purpose is. proof that $f'$ is continuous at $a$ Let $f:I \rightarrow \mathbb{R}$ derivable in set I. if $\lim x_n = \lim y_n =a \in I$ with $x_n \neq y_n$ and $\lim \frac{f(y_n)-f(x_n)}{y_n - x_n} = f'(a)$ proof that $f'$ is continuous at $a$  my attempt $x_n \neq y_n$ so by MVT $\exists z_n \in I$ like $x_n \geq z_n \geq y_n$ with $\frac{f(y_n)-f(x_n)}{y_n - x_n}=f'(z_n)$ for each $x_n$ and  $y_n$ then $f'(a) = \lim\frac{f(y_n)-f(x_n)}{y_n - x_n}= \lim f'(z_n)$   Using $\lim x_n = \lim y_n = a$ and $x_n \geq z_n \geq y_n \Rightarrow z_n \rightarrow a$  if $ \lim_{x\rightarrow a} f'(x) \neq f'(a)$ then $\exists \epsilon &gt; 0$ such $\forall n\in \mathbb{N}$, $z_n \in [x_n,y_n]$ $0 &lt; |z_n - a| &lt; \frac{1}{n}$ and $|f'(z_n) - f'(a)| \geq \epsilon$. So $z_n \rightarrow a$ but will do not have $\lim f'(z_n) = f'(a)$  so $\lim f'(z_n) = f'(a) = \lim_{x\rightarrow a} f'(x)$ when $z_n \rightarrow a$  and thats implies $f'$ continuous at $a$  my question is, that's correct ? if not, can you help me? thanks My problem is simple. I think that the strange deformations on the top are there because there are so many vertices in one face. How do I fix this while maintaining nice topology?  Although I triangulated the top polygon, the problem still persists. I want the figures and tables to be numbered 1, 2, 3, ... until the end of the document instead of 1.1, 1.2, ... in chapter I, 2.1, 2.2, ... in chapter II and so on.  I know how to convert the "chapter" part of figure/table numbers from Roman to arabic:  \renewcommand \thefigure     {\ifnum \c@chapter&gt;\z@ \@arabic\c@chapter.\fi \@arabic\c@figure} \renewcommand \thetable     {\ifnum \c@chapter&gt;\z@ \@arabic\c@chapter.\fi \@arabic\c@table}   One could modify the above code in order to remove the "chapter" part, but the figure/table counters would still be reset to 0 at the start of new chapters. Is full wave rectifier better than half wave one? Trig sum: $\tan ^21^\circ+\tan ^22^\circ+\cdots+\tan^2 89^\circ = \text{?}$ As the title suggests, I'm trying to find the sum $$\tan^21^\circ+\tan^2 2^\circ+\cdots+\tan^2 89^\circ$$ I'm looking for a solution that doesn't involve complex numbers, or any other advanced branch in maths. The solution can involve techniques such as induction, telecoping, etc, but preferably only ideas from precalculus, e.g. trig identities, polynomials, etc.  EDIT: I know that the sum is a rational number. bounding a $C^1$ function that upholds $f'(t)\leq 1-f(t)^2$ Maximum weight an average bicycle can carry? How to list all nodes of nodetype A which has the current node referenced? So this is what I'm trying to do in Drupal 6:  Node type A is video, node type B is a person Nodetype A has a reference field where you can enter a reference to any Node from nodetype B  how can I create a block view which will be displayed on nodes from node type b, which will show all the nodes from node type A where the current node from node type B is referenced? php sort associative arrays by keys that those keys exist in other array Elmah: How to get JSON HTTP request body from error report are more efficient than ordinary upright bicycles, and  are even more efficient. Then why do we see them so rarely on the road? Is it because they are expensive, or are there some non-financial, significant disadvantages? JavaScript variable definition: Commas vs. Semicolons As the question title suggests, how do I see that every left ideal of a square matrix ring over a field is principal? I received a table template, which I'm intended to generate dynamically with data using Java. But when the table becomes to large, the compilation with LaTeX fails. The specific about this template is, that it stretches the table width into the maximal page width and has same column widths for all data columns (except the first column).  Here is a minimal version of the full template, that still works fine:  \documentclass[final,authoryear,3p,twocolumn]{elsarticle}  \usepackage{amssymb,amsmath} \usepackage{calc} \usepackage{array} \newcolumntype{L}[1]{&gt;{\raggedright\arraybackslash}p{#1}} \newcolumntype{R}[1]{&gt;{\raggedleft\arraybackslash}p{#1}} \usepackage{eurosym} \usepackage{microtype} \usepackage{needspace} \usepackage{fp}  \usepackage{graphicx} \DeclareGraphicsExtensions{.pdf} \input{pgfexternal}  \newcommand\tabt{\rule{0pt}{2.6ex}} \newcommand\tabb{\rule[-1.2ex]{0pt}{0pt}}  \begin{document}  \begin{table*}[t]     \footnotesize     \begin{tabular*}{\linewidth}{@{}L{2em}@{}*9{R{(\linewidth-2em)/9}@{}}}\hline\tabt $i$ &amp; $1$ &amp; $2$ &amp; $3$ &amp; $4$ &amp; $5$ &amp; $6$ &amp; $7$ &amp; $8$ &amp; $9$\tabb\tabularnewline\hline\tabt $1$ &amp;  &amp; $48.87$ &amp; $0.01$ &amp; $0.05$ &amp; $1.23$ &amp; $0.00$ &amp; $0.20$ &amp; $5.67$ &amp; $2.35$\tabularnewline     \end{tabular*} \end{table*}  \end{document}   But when extending this example by another column, so that the table would look like the following example, it fails:  \begin{table*}[t]     \footnotesize     \begin{tabular*}{\linewidth}{@{}L{2em}@{}*10{R{(\linewidth-2em)/10}@{}}}\hline\tabt $i$ &amp; $1$ &amp; $2$ &amp; $3$ &amp; $4$ &amp; $5$ &amp; $6$ &amp; $7$ &amp; $8$ &amp; $9$ &amp; a\tabb\tabularnewline\hline\tabt $1$ &amp;  &amp; $48.87$ &amp; $0.01$ &amp; $0.05$ &amp; $1.23$ &amp; $0.00$ &amp; $0.20$ &amp; $5.67$ &amp; $2.35$ &amp; b\tabularnewline     \end{tabular*} \end{table*}   This fails, no matter how many columns I add. So far, it seems for me to be a problem with the division by 10. May it be, that LaTeX isn't able to handle that properly by default? How can I solve this problem and make even larger tables work using this template. Or is that to any other problem I don't see yet? Thanks for any help! There are lots (an infinitude) of smooth functions that coincide with $f(n)=n!$ on the integers. Is there a simple reason why Euler's Gamma function $\Gamma (z) = \int_0^\infty t^{z-1} e^{-t} dt$ is the "best"?  In particular, I'm looking for reasons that I can explain to first-year calculus students.