quaternions

In Python, complex numbers are typed as follows: 3+4j. This kinda bugs me that they use "j" instead of the traditional "i", particularly because quaternions (4-dimensional complex numbers) use i, j, and k for the three "imaginary" directions, so this would cause some overlap. But then I got to...

Okay, we all know (I presume) that we can "extend" the complex number system by the quaternion system of numbers. It is possible to take complex powers of complex numbers: (1 + i)^{1 - 2i} = \sqrt{2} e^{\pi / 2} ~ cos \left ( \frac{\pi}{4} - ln(2) \right ) + i \sqrt{2} ~ sin \left (...

I've been watching a set of videos on medical image processing, and I'm working through a section on using Quaternions for 3 d image registration. That is, given 2 sets of 3 points in R3, where one set has been rotated about an axis by an unknown amount, find the quaternion that best describes...

I got this question in my Group Theory class and I'm not really sure how to go about it because I never really understood quaternions in the first place! Let v be the unit vector ( $\frac{1}{\sqrt{3}}$,$-\frac{1}{\sqrt{3}}$,$\frac{1}{\sqrt{3}}$) and let R be the rotation through angle...

I've a problem from my Group theory module that's been puzzling me, which is asking me to rotate a vector x (1,0,0) around a vector Ov, where v is (1/sqrt(3), -1/sqrt(3), 1/sqrt(3)), by 60 degrees by representing x as a matrix Z (encoding (0,1,0,0) as a matrix) and computing UZU*, where U is...

Let G be the group {±1, ±i, ±j, ±k} where i^2=j^2=k^2=–1, –i=(–1)i, 1^2=(–1)2=1, ij=–ji=k, jk=–kj=i, and ki=–ik=j. (Quaternions group multiplication). Prove that G is isomorphic to Q = <a, b|a^4=e, b^2=a^2, and ab=b(a^3)>.

Prove that the quaternions \bold Q and the Dihedral group D_8 are non-isomorphic groups of order 8.

I have this function I need to build a Jacobian matrix for: F = R\left(\bar{q}\right) . p where p is a position vectors (x,y,z) and R(q) is the function converting a quaternion to a rotation matrix. Note that in the function F, the rotation matrix is obtained from the conjugate of q, i.e...

Hamilton's fundamental equations are: i squared =j sq=k sq=ijk=-1 the first part is by definition. my question is: can the second part ie ( i j k = -1) be derived from the first part? If so, how. I obviously have trouble!! many thanks for enlightening this old codger.

Prove that X^2+1 has infinite roots in the quaternions ring Thanks!!!

Before we go into math this is the common life explanation of the problem that I try to simulate: I have a piece of a tape (thin, wide, and long). One end of it is attached to a surface with a pin (tape can rotate around the pin). I can move the other end of the tape around as I wish. I'm using...