Quaternions Definition and 77 Threads

  1. M

    Exploring the Consequences of Galois Group Isomorphism to Quaternions

    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...
  2. S

    Before Vectors, was it Components, and Quaternions?

    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...
  3. N

    Angular velocity vs Quaternions

    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...
  4. H

    Quaternions, how to prove q^** = q

    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
  5. K

    Can Quaternions Revolutionize Quantum Mechanics?

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

    Conjugate and Inverse Quaternions

    Does an inverse exist for every quaternion under multiplication??
  7. G

    Quaternions or Vectors: Which is the Better Choice for Electromagnetism?

    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?
  8. M

    Quaternions and the ather hypercomplexe numbers

    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?
  9. P

    Prove that quaternions are associative

    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...
  10. F

    Quaternion Multiplication: Expanding and Simplifying

    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...
  11. L

    Coding in Quaternions and Matrices

    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...
  12. D

    How do Hypercomplex Numbers and Quaternions Work in Supersymmetry?

    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...
  13. P

    Understanding the Role of Quaternions: Algebra, Normed Linear Space, and Field

    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...
  14. E

    Correctly combining two Quaternions

    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...
  15. P

    How Does Quaternion Rotation Affect Another Quaternion?

    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?
  16. H_man

    Have I understood Quaternions correctly?

    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...
  17. R

    Anyone have a pdf of Maxwell's EQs in quaternions

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

    How Many Angles Can a Quaternion Represent in 3D Space?

    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...
  19. O

    Quaternions and hypercomplex numbers are incompatible

    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...
  20. A

    Quaternionic powers of quaternions?

    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?
  21. R

    What are the practical applications of quaternions in physics?

    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?
  22. K

    Quaternions And Complex Numbers

    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 ?
  23. quasar987

    Complex, quaternions, octonions,

    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:
  24. marlon

    Quaternions & Octonions: Definition & Uses

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

    Can quaternions be used to simplify Maxwell's equations?

    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 +...
  26. A

    What Are Quaternions and Their Relation to Special Relativity?

    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}...
  27. enigma

    I need to learn what quaternions are

    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...
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