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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Is the set of quaternions $\mathbb{H}$ algebraically closed?

A skew field $K$ is said to be algebraically closed if it contains a root for every non-constant polynomial in $K[x]$. I know that this is true for $\mathbb{C}$, which is the algebraic closure of $\mathbb{R}$ and a true field. I wonder if this is…
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Prove that the map $\phi:S^3\times S^3\to{\bf GL}(4,\Bbb R)$ defined via quaternions as$\phi(p,q)(v)=pvq^{-1}$ has image ${\bf SO}(4)$

I am interested in the map $\phi:S^3 \times S^3 \to GL_4(\mathbb{R})$ given as follows: Let $(p,q) \in S^3 \times S^3$. We identify $p$ and $q$ as real quaternions with unit norms and define $\phi(p,q)$ to be the map sending $v \in \mathbb{H}$ to…
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Can rotations in 4D be given an explicit matrix form?

Rotation in 2D by an angle $t$ can be performed using $$R=\begin{pmatrix}\cos(t) &-\sin(t) \\ \sin(t) & \cos(t)\end{pmatrix}$$ matrix. But, if I want to rotate a point or vector in 4D, is there any rotation matrix in explicit form? I have read…
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How to rotate one vector about another?

Brief Given 2 non-parallel vectors: a and b, is there any way by which I may rotate a about b such that b acts as the axis about which a is rotating? Question Given: vector a and b To find: vector c where c is produced by rotating a about b by an…
reubenjohn
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Quaternions: why does ijk = -1 and ij=k and -ji=k

Currently i am studying quaternions. I do understand that i, j and k are imaginairy numbers. so $i^2 = j^2 =k^2 = -1$. But I could not understand this: $$\begin{matrix}ij=k,&ji=-k,\\jk=i,&kj=-i,\\ki=j,&ik=-j\end{matrix}$$ Why is this? There seems…
Clifford
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Why are properties lost in the Cayley–Dickson construction?

Motivating question: What lies beyond the Sedenions? I'm aware that one can construct a hierarchy of number systems via the Cayley–Dickson process: $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S} \subset…
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Why don't quaternions contradict the Fundamental Theorem of Algebra?

I don't pretend to know anything much about the Fundamental Theorem of Algebra (FTA), but I do know what it states: for any polynomial with degree $n$, there are exactly $n$ solutions (roots). Well, when it comes to quaternions, apparently…
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Euler angles and gimbal lock

Can someone show mathematically how gimbal lock happens when doing matrix rotation with Euler angles for yaw, pitch, roll? I'm having a hard time understanding what is going on even after reading several articles on Google. Is the only work-around…
Xavier
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Composition of two axis-angle rotations

Please note that I am not referring to Euler angles of the form (α,β,γ). I am referring to the axis-angle representation, in which a unit vector indicates the direction axis of a rotation and a scalar the magnitude of the rotation. Let…
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Is there an algebraic closure for the quaternions?

This post is a sequel of: Is the set of quaternions $\mathbb{H}$ algebraically closed? This answer shows that: 1. $\mathbb{H}$ is algebraically closed for the polynomials of the form $\sum a_r x^r$ 2. It is not for the polynomials freely…
Sebastien Palcoux
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What does multiplication of two quaternions give?

I'm using quaternions as a means to rotate an object in the application I'm developing. If one quaternion represents a rotation and the second quaternion represents another rotation, what does their multiplication represent? Many web sites talk…
Nazerke
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Automorphism group of the quaternion group

Let $Q_8$ be the quaternion group. How do we determine the automorphism group ${\rm Aut}(Q_8)$ of $Q_8$ algebraically? I searched for this problem on internet. I found some geometric proofs that ${\rm Aut}(Q_8)$ is isomorphic to the rotation group…
Makoto Kato
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what is the relation between quaternions and imaginary numbers?

I understand the idea behind complex and imaginary numbers. I am trying to understand quaternions. What is the relation between imaginary (or complex) numbers, and quaternions?
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How to raise a number to a quaternion power

Now, I know that it's (relatively) easy to calculate, say, $r^{a+bi}$ (using the fact that, for $z_1, z_2\in \mathbb{C}, {z_1}^{z_2}=e^{z_2\ln(z_1)}$ and $\ln(z_1$) can just be found using: $\ln(a+bi)=\ln[\sqrt{a^2+b^2}]+i \cdot…
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If there are any 3nion, 5nion, 7nion, 9nion, 10nion, etc.

The quaternion/octonion extend the complex numbers, which extend the real numbers. So we go: 1-tuple: Real numbers. 2-tuple: Complex numbers. 4-tuple: Quaternions. 8-tuple: Octonions. The Wikipedia link describes this doubling process: In…
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