Find the unit vector $\vec{b}$ if $-\hat{i}+\hat{j}-\hat{k}$ bisects the angle between $\vec{b}$ and $\vec a=3\hat{i}+4\hat{j}$  The unit vector along the angle bisector is $$\frac{-\hat{i}+\hat{j}-\hat{k}}{\sqrt3}=\frac{\vec{b}+\left(\frac35\hat{i}+\frac45\hat{j}\right)}{|\vec{b}+\frac35\hat{i}+\frac45\hat{j}|}$$ Im not able to extract $\vec b$ from here
There are two vectors called $\vec{a}$ and $\vec{b}$. Vector $\vec{c}$ is the bisector and it can be given as  $$\vec{c} = |b|\vec{a} + |a|\vec{b}$$  How to prove that? I have used the dot product method. But there I can't find the angle between them. Then I tried to do this by using unit vectors. Then I got an answer like in the given picture. . Can you help me to get the correct answer
How do I create an invisible character? I cannot remember how to create an invisible character in LaTeX, i.e. I want to put a space that has the width of a particular character, say `M'. I know there is a command for this, and this is a really dumb question, but my Google-fu has failed me. Implementing support for nested transactions using JDBC 3.0 our legacy application uses JDBC 3.0. it supports transactions by implementing its own transaction manager that is capable of returning the same JDBC connection for each thread. The problem I recently discovered is that it doesn't support nested transactions: if a transaction is started inside an another transaction, then every SQLs running in the context of the inner transaction will be executed using the same db connection, and when it's commited or rolled back it will automatically commit or roll back all the changes starting from the outer transaction.  As far as I understand I can implement the support for nested transaction using JDBC 3.0 savepoints: whenever a nested transaction is started, I can set a new savepoint for the current connection. Afterward, if the nested transaction is rolled back, I will just rollback to this savepoint. If, on the other hand, it is commited, I will just do nothing. Only the commit of the most outer transaction will save the changes to the db.  Is this correct? Does this approach have any flaws? If yes, what are my possibilities?  Thanks. Prove that $|f''(\xi)|\geqslant\frac{4|f(a)-f(b)|}{(b-a)^2}$ Let ${\rm f}:\left[a, b\right]\to\mathbb{R}$ be twice differentiable, and suppose   $$\lim_{x\to a^{+}} \frac{{\rm f}\left(x\right) - {\rm f}\left(a\right)}{x - a} = \lim_{x\to b^{-}}\frac{{\rm f}\left(x\right) - {\rm f}\left(b\right)}{x - b} =0 $$  Show that there exists $\xi \in \left(a, b\right)$ such that $\displaystyle{% \left\vert\vphantom{\Large A}\,{\rm f}''\left(\xi\right)\right\vert \geq \frac{4\left\vert\vphantom{\Large A}% \,{\rm f}\left(a\right) - {\rm f}\left(b\right)\right\vert} {\left(b - a\right)^{2}}}$.  I don't know how to start. Any hints ?. Search question and answer Hyperlinks for a hostname? JSON encode MySQL results How do I use the json_encode() function with MySQL query results? Do I need to iterate through the rows or can I just apply it to the entire results object? Simple homework question about normally distributed variables The question states:      Consider a set of random variables $X_i$, where $i=1,...n$. Each $X_i$ is   normally distributed with mean $0$ and variance $1$, i.e. $X_i$ are $\mathcal N(0,1)$.   What is the mean and the variance of the random variable $Y$, where   $Y=X_1+...+X_n$.   How do I do this? Combinatorics Distribution - Number of integer solutions Concept Explanation I reading my textbook and I don't understand the concept of distributions or number of solutions to an equation.  It's explained that this problem is 1/4 types of sampling/distributions problems.  An example is provided to illustrate:     In how many ways can 4 identical jobs (indistinguishable balls) be distributed among 26 members (urns) without exclusion (since one member can do multiple jobs)?      A sample outcome might be:   $\text{_____________}$   |  A  |  B  |  C  |...|  Z  |    $\text{--------------------}$      $\text{_____________}$   |  o  | oo|    |   |    ||| o  |    $\text{_____________}$   |  A  |  B  |  C  |...  |  Z  |    $\text{--------------------}$      Thus, the question is reduced to, "How many $(26-1+4)$ letter words are there consisting of four circles and $(26-1)$ vertical lines?"  Therefore, the solution is:  $\binom{26-1+4}{4}$   I really don't understand why its $(26-1+4)$.  There's only 26 different spots or "urns" to place the 4 jobs.  Can someone please explain?    I looked through another text to try and understand and I found it explained as such:       There are $\binom{n+r-1}{r-1}$ distinct nonnegative integer valued vectors $(x_1,...,x_r$ satisfying the equation $(x_1 + ... + x_r = x_n$ for $x\ge0$. $\spadesuit$   How in the world are they deriving this?  For distinct positive integers I understand:  Assume I have 8 balls (n=8) and I have 3 urns (r=3): o^o^o^o^o^o^o^o, where o represents  a ball and ^ represents a place holder where an urn could be placed.  For this scenario:     There are $\binom{n-1}{r-1}$ distinct positive integer valued vectors $(x_1,...,x_n)$ satisfying the equation:  $x_1 + ... + x_n = n, x_i&gt;0, i=1,..,r$ $\clubsuit$   It's clear that I could have this specific case ooo|ooo|oo.  Here the bar represents a divide for the urn and you see I have 3 sections.  So that case is clear.  Can anyone please explain this problem to me?  I don't understand the nonnegative integer case.   Also, people who post tend to be crazy smart and explain things very in a complicated manner.  I'd appreciate it if it could be explained in layman's terms as much as possible.   Thank you!!! Is “of ” necessary in “all of ”? Listen to all your fans   Name all the states   vs      Listen to all of  your fans   Name all of  the states   What part of language is of  in these examples? Is it necessary or optional, correct or incorrect? The topic which haunted me for two years until I gave up on it. But now I am doing engineering and this topic suddenly popped out of my textbook from nowhere. I seriously need to understand this topic because I wasn't able to do so in the past. I have read many book on 'energy' and it came to nothing. Maybe just because the many books I have read all said 'energy' is something which doesn't 'exist'! It is 'abstract'! It is just a number which represents the state or orientation of a system. But then I see so many examples that 'uses' energy to do 'work'. But the question here is that if anything doesn't exist in this universe, then how can the same thing be used to do something that exist?  My problem is that I didn't understood the topic 'energy' and all the other topics related to it (work, power etc.) I'd like to know if there is any alternative to Deep Freeze for Linux that still under development, because I've read some things on the internet and looks like the same company which developed Deep Freeze had a Linux version of it, but the project was discontinued. Minimal cut edges number in connected Eulerian graph. I have to prove that Connected graph is Eulerian Graph if and only if the number of edges in minimal cut is even.  Implication from left to right. Well if $G$ is Eulerian Graph then degree of each vertex must be even. It's oblivious if i try to disconnect graph by removing all edges connected to the vertex with minimal degree. But minimal cut can be done also by dividing graph into 2 connected graphs (each graph with more then one vertex) and i have no idea how to deal with it.  Implication from right to left - completly no idea. I'm trying to convert some strings that are in French Canadian and basically, I'd like to be able to take out the French accent marks in the letters while keeping the letter. (E.g. convert é to e, so crème brûlée would become creme brulee)  What is the best method for achieving this? Analogue of spherical coordinates in $n$-dimensions What's the analogue to spherical coordinates in $n$-dimensions? For example, for $n=2$ the analogue are polar coordinates $r,\theta$, which are related to the Cartesian coordinates $x_1,x_2$ by  $$x_1=r \cos  \theta$$ $$x_2=r \sin  \theta$$  For $n=3$, the analogue would be the ordinary spherical coordinates $r,\theta ,\varphi$, related to the Cartesian coordinates $x_1,x_2,x_3$ by  $$x_1=r \sin  \theta  \cos  \varphi$$ $$x_2=r \sin  \theta  \sin  \varphi$$ $$x_3=r \cos  \theta$$  So these are my questions: Is there an analogue, or several, to spherical coordinates in $n$-dimensions for $n&gt;3$? If there are such analogues, what are they and how are they related to the Cartesian coordinates? Thanks. I can't see anything - Teams banner and cookie toaster obscure my whole screen There have been  for a while, but this is ridiculous. It reminds me of , but this time it's network-wide.   The banner must be scrolled past (or dismissed) on every page unless both first-party JavaScript and first-party cookies are enabled. If first-party JavaScript is enabled, and the banner is visible, the rest of the page jumps up and down every so often.  Under certain circumstances (not quite sure what they are, yet – a direct link to an answer, maybe? I'm seeing it ) the entire page jumps up and down a line every few seconds even when the banner is off-screen.   The cookie banner keeps coming back unless both third-party JavaScript  are enabled (and  is solved). The cookie box occludes the “Learn more about Teams” link in the banner (with default Firefox window sizes) and the Job / Teams link in the sidebar.  As it stands,  – a site for hobbyists and researchers, whose audience is not very likely to purchase Teams – is much, much harder to use, for no benefit that I can see. Can you roll back this change, please? Backtrack, Wi-Fi not working I've installed Backtrack 5R3 KDE, and I realized that my wireless is not working, but wired is working fine.  Here's the lshw output:     *-network         description: Ethernet interface         product: RTL8101E/RTL8102E PCI Express Fast Ethernet controller         vendor: Realtek Semiconductor Co., Ltd.         physical id: 0         bus info: pci@0000:02:00.0         logical name: eth0         version: 05         serial: 04:7d:7b:b7:46:f8         size: 100MB/s         capacity: 100MB/s         width: 64 bits         clock: 33MHz         capabilities: pm msi pciexpress msix vpd bus_master cap_list ethernet physical tp mii 10bt 10bt-fd 100bt 100bt-fd autonegotiation         configuration: autonegotiation=on broadcast=yes driver=r8169 driverversion=2.3LK-NAPI duplex=full firmware=rtl_nic/rtl8105e-1.fw ip=192.168.2.2 latency=0 link=yes multicast=yes port=MII speed=100MB/s         resources: irq:42 ioport:2000(size=256) memory:f0404000-f0404fff memory:f0400000-f0403fff   lspci output:      00:00.0 Host bridge: Intel Corporation 2nd Generation Core Processor Family DRAM Controller (rev 09) 00:02.0 VGA compatible controller: Intel Corporation 2nd Generation Core Processor Family Integrated Graphics Controller (rev 09) 00:14.0 USB Controller: Intel Corporation Panther Point USB xHCI Host Controller (rev 04) 00:16.0 Communication controller: Intel Corporation Panther Point MEI Controller #1 (rev 04) 00:1a.0 USB Controller: Intel Corporation Panther Point USB Enhanced Host Controller #2 (rev 04) 00:1b.0 Audio device: Intel Corporation Panther Point High Definition Audio Controller (rev 04) 00:1c.0 PCI bridge: Intel Corporation Panther Point PCI Express Root Port 1 (rev c4) 00:1c.1 PCI bridge: Intel Corporation Panther Point PCI Express Root Port 2 (rev c4) 00:1d.0 USB Controller: Intel Corporation Panther Point USB Enhanced Host Controller #1 (rev 04) 00:1f.0 ISA bridge: Intel Corporation Panther Point LPC Controller (rev 04) 00:1f.2 SATA controller: Intel Corporation Panther Point 6 port SATA AHCI Controller (rev 04) 00:1f.3 SMBus: Intel Corporation Panther Point SMBus Controller (rev 04) 02:00.0 Ethernet controller: Realtek Semiconductor Co., Ltd. RTL8101E/RTL8102E PCI Express Fast Ethernet controller (rev 05) What is the best technique for long-exposure photography in good light? How do I calculate someone's age based on a DateTime type birthday? Pushforward of inverse map at the identity? Let $G$ be a Lie group and $i:G \rightarrow G$ denote the inversion map $i(x) = x^{-1}$. (Notation: $f_*$ is the pushforward map $F_*:T_pG \rightarrow T_{i(p)}G$ which takes $(F_{*}X)(f)=X(f\circ F)$ and $X$ is a tangent vector, $X\in T_pG$.) I wish to show that $i_{*}:T_{e}G\rightarrow T_{e}G$  is given by $i_{*}(X)=-X$ As a first step, it is trivial to prove that $i_*$ is an involution as $\mbox{Id}_{*}=(i\circ i)_{*}=i_{*}\circ i_{*}$ but I can't seem to make any further progress. Any help would be appreciated. Is there a way to have a copy of any message sent from a Gmail account sent to another address?  I've got any messages coming to my address copied to another (see  for one solution, setting up filters is another), but I haven't yet found a way to have outbound messages automatically copied.  Is there a way? How do I make an object face the mouse position, but only on one plane? I want a game object to turn and look towards where the mouse was clicked (in world space).  It works well when considering all 3 dimensions, but I really only care for the plane formed by the x and z axis, not the y.  I've tried to fix this by setting the y of the directionTarget to the y of the object's position, but that gives weird results.  My code looks like this:  Quaternion rotationTarget;     float rotationLerpProgress;     float rotationLerpDuration = 1f;      // Use this for initialization     void Start ()     {      }      // Update is called once per frame     void Update ()     {             if (Input.GetMouseButtonDown (0)) {                     var ray = Camera.main.ScreenPointToRay (Input.mousePosition);                     RaycastHit rayHit;                      if (Physics.Raycast (ray, out rayHit)) {                             var rayHitPoint = rayHit.point;                              var rotationDirection = transform.position - rayHitPoint;                                                            rotationDirection.Normalize ();                             //rotationDirection.y = transform.position.y;                             rotationTarget = Quaternion.LookRotation (rotationDirection);                             rotationLerpProgress = Time.deltaTime;                     }                     //targetPosition = Camera.main.ScreenToWorldPoint (Input.mousePosition);                                     }              if (rotationTarget != transform.rotation) {                      transform.rotation = Quaternion.Lerp (transform.rotation, rotationTarget, rotationLerpProgress / rotationLerpDuration);                     rotationLerpProgress += Time.deltaTime;                       if (rotationLerpProgress &gt;= rotationLerpDuration) {                              transform.rotation = rotationTarget;                     }              }      }   I must be missing something. How do I do this right?