Questions tagged [quaternions]

For questions about the quaternions: a noncommutative four dimensional division algebra over the real numbers. Also for questions about quaternion algebras.

The ring of quaternions is a four dimensional division algebra over the real numbers. They are usually denoted as $\Bbb H$ in honor of the discoverer, William Rowan Hamilton.

The construction of the quaternions was given by Hamilton as follows: take three symbols $i,j,k$ and define $i^2=j^2=k^2=ijk=-1$. As a result, $ij=k$, and $jk=i$ and $ki=j$. Furthermore, $ji=-k$ and $kj=-i$ and $ik=-j$, so $kji=1$.

A quaternion is a linear combination $q=\alpha+\beta i+\gamma j +\delta k$ where $\alpha, \beta,\gamma,\delta\in \Bbb R$. Multiplication between quaternions is carried out by using the distributive rule and the rules for $i,j$ and $k$.

The quaternions turn out to be a noncommutative division ring. In fact, $\Bbb R$ and $\Bbb C$ and $\Bbb H$ are the only associative finite dimensional division rings over $\Bbb R$. They are also the only normed division algebras over $\Bbb R$.

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Quaternion relation proofs (e.g.: $ik=-j$)

How do you prove that these relations are correct $(ij = k, jk = i, \ldots)$? I tried to prove some of them, and I could, but for example: ik = -j -j = -1 * j = (ijk) * j = i*(j^2)*k = -1 (ik) = -ik so -j = -ik... which is wrong, ** in which…
Snowman
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Conjugation of quaternions

This proof is extreeeeemely boring, but still I must get it right. Let $x = x_0 + x_1 i + x_2 j + x_3 k \in \mathbb{H}$ (the Hamilton quaternions). Conjugation is defined as: $$x^\ast = x_0 - x_1 i - x_2 j - x_3 k \, .$$ Show that for any $x, y\in…
Ystar
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Velocity vector transformations with respect to a global frame of reference

I have an object moving in a 3D space. This space has three coordinate axes and these are global axes, and space is the global reference frame. The object also has three coordinate axes and these can be in any orientation in respect to the global…
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Align the cube's nearest face to the camera

I have a cube and 4x4 transformation matrix Cube is rotated randomly I need to find the nearest face of cube regarding to camera and rotate the cube by aligning that face to the camera. How can I do that. Thanks in advance.
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Question regarding practical SLERP

We are suppose to compute the quaternion which performs 1/5 of the rotation of this quaternion: [ 0.965 (0.149 -0.149 0.149)] The answer provided is shown as below: Picture of the answer I'm stuck at the second last row of the ans. How do you…
p0larBoy
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Find all Quaternions Satisfying..

Let H be the skew field of quaternions. Find all quaternions x satisfying $(i + j)x(i + k) = 2$ I'm having trouble figuring out what to do with this question. I know the "i j k i j k" formula for determining the cross-product of vectors. $(ij = k,…
Squires McGee
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Rotation Equivalence using Quaternions

I'm given a statement to prove: A rotation of π/2 around the z-axis, followed by a rotation of π/2 around the x-axis = A rotation of 2π/3 around (1,1,1) Where z-axis is the unit vector (0,0,1) and x-axis is the unit vector (1,0,0). I want to prove…
DashControl
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Quaternion Negative Unity

I'm reading Hamilton's Paper on Quaternions. Found here http://www.emis.de/classics/Hamilton/OnQuat.pdf. On page 5, the first statement of 7, says that there are only two different square roots of negative unity. I tried googling and I can't find…
Gakho
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Question about Hamilton's Quaternion Paper

So I was reading Hamilton's paper on quaternions. http://www.emis.de/classics/Hamilton/OnQuat.pdf. On page 2, I'm having trouble following how QQ' and equations A,B,C lead to equation D. My main question is in the original expression for QQ', we…
Gakho
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Basis for quaternionic functions

We know that the set of functions $\{1,\cos x, \sin x, \cos 2x, \sin 2x, ... \; | \,x \in \mathbb{R} \}$ is a basis in the space $L^2_\mathbb{R}[-\pi,\pi]$ . Given a quaternion $z \in \mathbb{H}$ we can analogously define the functions $$ \cos…
Emilio Novati
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how to offset a rotation contained in a unit quaternion rotation from the origin of a rigid object.

I'm using Unity3D for a project. The way it handles sorting transformations is with a 3vector-unit quaternion-3vector "sandwich" (the 1st vector for position, the quaternion for rotation, and the 2nd vector for uniform axis scale). When you add a…
RhinoPak
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What is the theory for the quaternions?

The axiomatic theory for the natural numbers is PA. For the reals it's RCF and for the complex numbers it's ACF. What's the theory for the quaternions?
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Tricks in Complex analysis, but not possible in Real analysis or Quaternion analysis

What are some fundamental reasons that Complex analysis in $\mathbb{C}$ is more powerful, but it is not possible to generalize to the Real analysis in $\mathbb{R}$ or the Quaternion analysis in $\mathbb{H}$? What makes the $\mathbb{C}$ more special…
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Angle of rotation by pure unit quaternions(fundamental unit quaternions)

From the versor definition here, i understand that versor is a unit quaternion and also a pure quaternion with its scalar part zero. So, versor is a pure unit quaternion. I think that fundamental quaternion units $i,j,k$ are also versors(!). From…
lockedscope
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What is the "unit axis"?

As the highlighted text in the screenshot, what is the unit axis? And what mathematical properties the three coordinates X, Y, Z have?
niebayes
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