i trying do some revise for my test. i have trouble convert rotation matrix to quaternion form example given this matrix from quaternion &lt;1,2,3,-5&gt;, generate from  [ -0.7435898,  0.5641026, -0.3589744 ] [  0.0512821, -0.4871795, -0.8717949 ] [ -0.6666667, -0.6666667,  0.3333333 ] what the step to get back the quaternion form?
We use unit length Quaternion to represent rotations. Following is a general rotation matrix obtained ${\begin{bmatrix}m_{00} &amp; m_{01}&amp;m_{02} \\ m_{10} &amp; m_{11}&amp;m_{12}\\  m_{20} &amp; m_{21}&amp;m_{22}\end{bmatrix}}_{3\times 3}\tag 1 $.   How do I accurately calculate quaternion $q = q_1i+q_2j+q_3k+q_4$for this matrix?Means   how can we write $q_i$s interms of given $m_{ij}$  accurately?
Improve alignment of "Learn more" button in "Newest badge" popup The popup when you click on your newest badge in your user profile has a &quot;Learn more&quot; button, but its alignment is suboptimal; it's too close to the top of its container:  It looks better with a 4px top margin: Automatically switch gpu graphics driver when gaming I have installed Ubuntu GNOME 15.10. I installed the nvidia tested proprietary driver and then I restarted. Afterr that I selected as primary GPU the Intel driver for normal use. However, when I enter a steam game(Portal) the graphics is bad so In order to play normal I have to switch manually to the nvidia driver again and reboot again. After the game I have to manually select as primary driver the Intel one and reboot again. It is so boring. QUESTION: How to automatically switch gpu graphics driver when I enter a game or something that needs more GPU? The following code best describes the situation.  Why is the last line not outputting the trailing newline char?  Each line's output is shown in the comment.  I'm using GNU bash, version 4.1.5          echo -n $'a\nb\n'                  | xxd -p  # 610a620a              x=$'a\nb\n'   ; echo -n "$x" | xxd -p  # 610a620a      echo -ne "a\nb\n"                  | xxd -p  # 610a620a x="$(echo -ne "a\nb\n")" ; echo -n "$x" | xxd -p  # 610a62 Memory management in Objective-C Spoilers for A Storm of Swords / Season 4 episode 2 "The Lion and the Rose".  At the Purple Wedding,      Littlefinger and Lady Olenna Tyrell   perform their plan of      poisoning King Joffrey.   A short summary of their plan:  Littlefinger orders Ser Dontos Hollard to give Sansa Stark a beautiful silver hairnet adorned with amethysts. At least some of the amethysts are actually strangler crystals in disguise. Ser Dontos also, still under orders of Littlefinger, urges Sansa to wear the hairnet at the wedding, which she then does. At the wedding, when Lady Olenna greets Sansa, she fusses her hairnet a bit, while secretly taking one of the strangler crystals. Later, during the wedding, Lady Olenna puts the crystal in Joffrey's royal cup, poisoning him.  But why was their plan so complicated? Why couldn't Lady Olenna just bring a strangler crystal with her to the wedding (as suggested in )?  Then they wouldn't depend on Ser Dontos and Sansa who could unknowingly have done something to prevent the success of the plan.  The only reason that I could think of, is Littlefinger and Lady Olenna plan for Sansa (and Tyrion as her husband with her) to take the blame for Joffrey's death.  They put the crystals on her person, so that upon examination it would clearly show that Sansa had the murder weapon on her. But Littlefinger had a plan he did not tell Lady Olenna, namely rescuing Sansa during the chaos ensuing the murder (and thereby leaving Tyrion behind as the only accused).  Is there anything confirming or debunking this theory? If my theory doesn't hold up, is there another explanation for why the plan was so needlessly complicated? What figure of speech is this? Why is there free space at the end of NTFS Partitioned Drives? I've been wondering this for a while, why is there free space at the end of a drive that was formatted to NTFS in windows?  You can only see the unallocated space in Linux. So why is it there? Does it serve a purpose? How does a wheelie work? So I've been trying to create a mathematical model for an electric motorcycle and began to wonder about the maximum possible torque that could be supplied to the rear driven wheel without having the bike begin to lift up on one wheel.  I found ways to calculate this value online; however, the basic concept as to how the bike actually lifts escapes me.    My problem started when I began to think about what axis the bike will rotate about when it is doing the wheelie.  My first intuition was that the frame and front wheel, together, rotate about the rear axle.  But when I drew a free body diagram of the frame/front wheel system just at liftoff (see below) I made note that the only forces acting about the rear axle, point O, is the force of weight.  This means that an increase in applied torque and subsequently, the applied force Fa, should not effect the rotation about the rear axle.  I know that the applied force on the back wheel is indeed correlated with the propensity for a bike to wheelie, so I considered that the axis of rotation I was choosing was wrong.  If we take the free body diagram above and sum the moments about the center of mass, we would find that an increased applied force would in fact cause the solid body to rotate.  The problem with this understanding is that during rotation, the center of mass of the drawn system should actually rises relative to the surface the bike is moving across.  If the bike were to be truly rotating about its center of mass, then the back wheel would begin to dip below the surface of the road like you might see in a glitchy video game.    So I suspect that the bike is in fact rotating about the back axle, but I don't understand why, please help!  edit: I added the external torque from the back wheel to the frame, which would allow the bike to rotate about the back axle  edit 2:  I suppose that the torque acting on the frame via the engine should not effect the rotation as the movement of a motorcycle can be perfectly replicated by applying a force at the back axle.  A third possible axis of rotation might be the lowest point of the back wheel. How can I get this symbol without using bbold? A solution I saw somewhere suggested to use   \usepackage{bbold} $\mathbb{1}$   which works well and I get what I want. However, now other symbols used to represent real numbers, natural numbers, etc look ugly.    I'd like them to look (without using bbold)    I'd like to only use that package to achieve that strange double one. Is there any way I can do that? Displaying previous and current year with dynamic text? How to upgrade to GNOME 3.30 on Ubuntu 18.04.1 How to fix a stripped pedal crank The threads on my bike's pedal crank just became fully stripped and the pedal fell off, though the pedal itself is fine. Is there any way I can fix this without fully replacing the pedal crank? I recently bought a Minolta film camera and it had some film with it. The film is dated 1991 - 1994. Can I use the film? Is Mana Weaving ok if it's followed by a thorough shuffle? My friends and I are relatively new to MtG. At some point, one of them introduced us to the idea of mana weaving, and we'd do this to declump our decks prior to a very thorough shuffling. However, at the Theros pre-release, one of us was asked not to do this, and we learned it was frowned upon and ceased to do it.  I just now came across , and it sounds like mana weaving's only bad if you use it to leave your deck in an even distribution, and it's OK if your deck is then thoroughly shuffled. In hindsight, the judge may have simply thought my friend wouldn't have shuffled afterwards, but at the time I came under the impression it was frowned upon even if you did shuffle afterwards.  So: is mana weaving my deck OK, if I shuffle it very thoroughly afterwards? When updating, I get the following error message:   W: The repository 'http://ppa.launchpad.net/mc3man/trusty-media/ubuntu xenial Release' does not have a Release file.   Here, I find another statement on this error:   This recommends removing certain PPAs; and, I'm not sure if I should do that since it might mean not getting the updates that I need.    Is this what I should do? How to add a matrix to a LaTeX document How do we add a matrix to a LaTeX document? I have 1:50 wound current transformer and I don't have the datasheet. Could anybody help me to calculate the burden resistor?    Maximum current to be measured: 2A  Max output voltage to ADC: 5V I am working on a website for a client, and even though the site ranks extremely well for various different keywords (which is good), if I search for the site's name the homepage does not appear even on the second page on Google. I have to add the .com so it will appear.  But, for big brands like Best Buy, Nike and millions of others it is enough to just search for the brand itself and the first or second suggestion on Google is the brand's homepage.  How can I raise my brand's authority in Google so it will appear in Google on the first page when some searches for it?  I have tried changing the breadcrumbs so instead of "HOME" it is now the brandname, linked to the homepage. I thought that it will help with internal linking if the anchor text is the brand's name instead of the word "HOME".  Any ideas please? Is the "the ... the ..." construction governed by some grammatical rule? a) The more I study, the less I learn. b) More I study, less I learn.  Could we rewrite the sentence shown under the letter a) in the form shown under the letter b) without breaking any grammatical rule? If not, why not? Alternatively, why does the force created by blowing out air feel so much stronger than the force created by sucking in air?  Ok, so forget the human factor involved in blowing out candles. Consider a vacuum cleaner with a suction end and a blower end. Anyone who has tried it out notices the blower end creates a much stronger force than the suction end, despite the discharge being (more or less) equal at both ends.  Why does this happen?