Quaternions Definition and 82 Threads

I am interested in using hypercomplex numbers and not using tensors. Therefore a question about the difference between spinors and vectors. I read that they both can be written as quaternions. Vector: Vq = ix + jy + kz Spinor: Sq = ix + jy + kz So what is the difference between...

All right, I've been doing a lot of reading on quaternions, and while I think I understand how to use them, I'm still very confused as to why certain things are defined the way that they are. First question: Why, when multiplying the imaginary portions of quaternions do you get these "rules"...

I've seen quaternions mentioned in a few articles online and I think they could be a very interesting subject. I would like to learn about them in simpler terms first. Can anyone give me the rundown on what they are and how they work?

This statement was made in my class and I'm trying still to piece together the details of it... We say that some rational polynomial, f has a Galois group isomorphic to the quaternions. We can then conclude that the polynomial has degree n \geq 8. I have a few thoughts on this and I might...

What did physicists use before the introduction of vectors by Gibbs & Heaviside, was it the exact same as we would use when denoting components with an x or y subscript or something completely crazy? Also, I've read in quite a few places that quaternions are very useful for things like Special...

I would be extremely grateful if somebody could help me out with the following setup. I have a robot arm of x chains, where each chain element, i, at time t, is defined by length l_i(t) and quaternion q_i(t) (where the quaternion is a rotation to rotate from a previous chain element to...

Hello everyone, I need some information with angular velocity in 3D and since I'm a math student, it's been a long time I haven't worked with physics, especially mechanics. This is why I need your help. Let's say I have this billiard ball (it's just an example) and I hit the ball with a cue...

Homework Statement quaternion q = a + bi + cj + dk conjugate q* = a - bi - cj - dk I do not know how I get (q*)* = q The Attempt at a Solution

Has anybody ever thought of using quaternions in QM? If so, why stop there? WHy not use octonions, etc. ? Just curious ...

Does an inverse exist for every quaternion under multiplication??

I read that people prefer vectors over quaternions (e.g. for electromagnetism), since one can do the same operations more transparent. Is it really a matter of preference? What's advantages of one or the other?

apologie me if this question isn't corret or is simple for you do solutions of an equation as quaternions or as the ather hypercomplex numbers exist? wath do they do in physics? for exemple ottonions or sedenions do ather hypercomplex numbers exist?

Homework Statement prove that quaternions are associative. ie (qr)s = q(rs), where q,r,s are quaternions This isn't really a HW problem since I'm just wondering if there's a simpler way to prove associativity than the method I tried below Homework Equations i^2 = j^2 = k^2 = -1...

Homework Statement Expand and simplify the product of two quaternions: (3 + 2i + 3j + 4k)(3 + 3i + 2j + 5k) Justify your response. The Attempt at a Solution I have done this by expanding brackets normally, keeping the ijk's in the same order because the multiplication is not...

Hi , I am currently using quaternions in my dynamics simulation library for the orientation of rigid bodies . The bodies are very simple spacecraft orbiting around Earth and what I need is the relative yaw pitch and roll of one spacecraft against an other one (let's call them chaser...

I am currently doing some research in Clifford Algebra. My topic is to find explicit representation of its basis through matrices with entries in clifford algebra itself. At this moment I am trying to write code in mathematica to do KroneckerProduct and Quaternion multiplication. Now I have...

Only recently I discovered that there is a class of complex numbers named hypercomplex numbers. Hamilton invented (or discovered) the quaternions in 1843. i² = j² = k² = ijk = -1 I understand how complex numbers work, but I don't understand how hypercomplex numbers work. Can someone...

I read in some text the following: Quaternion algebra becomes a normed vector(linear) space with appropriate norm ...(blah blah)... Also since every element has a multiplicative inverse it is a field. Now, what I find confusing is that according to above a mathematical object called...

Hi folks, I have a little problem which seems to be melting my brain. I have a (software) model which I wish to rotate based on the topography of the ground. I'm using two angles to represent the terrain - a "North" and "East" elevation (hopefully that's self explanatory). Once I have the...

If we take a vector "v" and utilize a quaternion q and its conjugate complex, we can rotate the "v" vector this way: qvq* The question is, what happens if "v" is not a vector, and is a quaternion? rotates it?

Have I understood Quaternions correctly? Hi all, I have been trying to understand how quaternions work when rotating coordinate axes in 3-D. I think I may have finally succeeded but I would appreciate if someone who really knows how to use them could confirm if I am correct. I think that...

Just curious. You can do a lot of stuff with quaternions that you can't in vectors.

Dear Friends, I've read about quaternions, and they can be expresed in 4 terms or in angles like this: a + ib + jc + kd = cos \theta + \vec{v} sin \theta Quaternions are a generalization of the complex numbers, but in 3D. My question is about angles. For example, with a complex...

Extending the number system from complex numbers, (a+bi), to 4-D hypercomplex numbers, (a+bi+cj+dk), leads to a multiplication table such as: (A) i^2=j^2=-1, ij=ji=k, k^2=+1, ik=ki=-j, jk=kj=-i. Note that these hypercomplex numbers are commutative and have elementary functions. We...

Can such things be defined? I know there are complex powers to complex numbers (using polar forms), but what about quaternions (or perhaps complex powers of quaternions...). Any ideas?

In my infinite curiosity, I am reading about quaternions and his algebra. I have readed that they are used principally in "pure mathmatics", but there are applications in 3D modelling, and others... there are applications to the physics?

1. Are the HAMILTON‘ian unit vectors i, j, k still valid beside the imaginary unit i(Sqrt(-1))? Can we expand quaternions using complex numbers? 2. Is the quaternion a+bi+0j+0k equal to the complex number a+bi ?

What are called the numbers with 15 imaginary part and 1 real? And is that the limit or are there people working with numbers of more than 15 imaginary parts? If so, how many, and what's the name for them? :smile:

Can anyone give me a definition of quaternions and octonions? What are these things and what are they used for. regards marlon

This thread is the sequel to my other thread, 'Quaternions and SR'. My goal is to write some physical equations with quaternions. Because I think quaternions represent the 4-dimensionality and metric used in special relativity. See: A quaternion is a generalized complex number: A = a_t +...

Hi all, just discovered the LaTeX feature, so why not play around with it a bit. Quaternions are (sort of) generalized complex numbers where you have one real unit and 3 imaginary units i, j, and k. The basic equation is i^2 = j^2 = k^2 = ijk = -1. Now, if we have a 4-vector \vec{r}...

Hi all, For a project, I need to learn what quaternions are, and how they are used and manipulated. All I've had in my classes on the subject is: "They are another way of using angles to keep track of orientation, but we won't be covering them here." Same in all my textbooks - they're...