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

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How does a quaternion differ from a position in terms of algebraic structure?

For two positions, I can subtract one from another to get a vector; I can take combination of them to get another position. My question is, can I treat quaternions in the same way? To be more ...
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1answer
127 views

Simple Vector Space Rotation

Given Data in the problem & notation convenstions We have 3 rotation vectors called $\vec{\theta_1},\vec{\theta_2},\vec{\theta_3}$, magnitude of these vectors will give you angle of rotation We ...
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2answers
352 views

Quaternion–Spinor relationship?

I've known for some time about the rotation group action of the ('pure') quaternions on $ \mathbf{R}^3 $ by conjugation. I've recently encountered spinors and notice similarities in their definitions ...
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1answer
133 views

Showing $\mathbb{H}$ is isomorphic to a subring of $M_2(\mathbb{C})$ as $\mathbb{R}$-algebras

I'm currently trying to show that the ring $A =\begin{pmatrix} a & b \\ - \bar{b} & \bar{a} \end{pmatrix} $ is isomorphic to the real quaternions $\mathbb{H}$ as $\mathbb{R}$-algebras. ...
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2answers
659 views

Is there a relationship between the cross product and quaternion multiplication?

I've just been introduced to the Kronecker delta, $\delta_{ij}$, along with the alternating tensor, $\varepsilon_{ijk}$ (in vector calculus). Motivation for the question: I've been introduced to ...
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0answers
145 views

3D rotation by a complex angle around a complex axis.

Context In electromagnetism, it is common to meet complex angles. Snell's law can produce them (with metals and/or total internal reflection), and when we plug them into Fresnel's equations, the ...
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1answer
84 views

Quaternions $\Leftrightarrow $ Rotations -Conceptual question

I have an angular velocity vector analogue defined as(called Darbaoux vector some times in curve) $ \omega(s)=\frac{\mathrm{d}\theta(s) }{\mathrm{d} s}=\delta\theta(s)_x \vec{i}+\delta\theta(s)_y ...
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Rotating Frames on Curve

We are describing a curve with a moving frame. Figure 1 says about the definition of curve Figure 1 Specifications and Proprieties of the curve We can make a rotation 3D Matrix $R(s)_{3 ...
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1answer
42 views

Rotation about an axis by matrix multiplication

Suppose I have three axis of rotation vectors $\vec{v_1},\vec{v_2},\vec{v_3}$ and angle of rotation as vectors $\theta_1,\theta_2,\theta_3$. Take a vector $P$ then apply rotation around $\vec{v_1}$ ...
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1answer
151 views

Quaternions as a Lie algebra, its derivations

Let $\mathbb{H}$ be the algebra of quaternions. It can be proven that each derivation $D:\mathbb{H}\to \mathbb{H}$ is inner that is of the form $\mathrm{ad}x$ for some $x\in \mathbb{H}$. I am to prove ...
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2answers
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Matrix exponentiation intuition.

What does $x^A$ intuitively mean if $x \in \mathbb{C}$ and $A$ is any matrix? Also, what if we had $x$ being a matrix too? Last but not least, what happens if we have a complex $x$ raised to a ...
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Quaternion Integration - And conversion to 3D matrix

I have a rotation matrix let us say $R(t)$ and its quaternion $q(t)$. We know already how to convert a quaternion to rotation matrix. Now if I want find $\int R(t) \ dt \tag1 $ can we do that in ...
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1answer
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Prove $\alpha i=i\alpha$ iff $c=d=0$. Let $\alpha \in \mathbb H$ and $\alpha=a+bi+cj+dk, a,b,c,d \in \mathbb Q$.

Prove $\alpha i=i\alpha$ iff $c=d=0$. Let $\alpha \in \mathbb H$ and $\alpha=a+bi+cj+dk, a,b,c,d \in \mathbb Q$. My Attempt: $(\rightarrow):$ $$\alpha i=ai-b-ck+dk \Rightarrow -b+ai+dj-ck$$ ...
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1answer
585 views

Quaternion , DCM , Euler Angles and Rotation Matrix Differences and when to use?

Quaternion , DCM[Direction Cosine Matrix] , Euler Angles and Rotation Matrix Differences and when to use ? All of the above components can represent rotation , so when to use each of them , best ...
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1answer
583 views

Reaching a point B in Cartesian coordinate via Euler angles knows its initial point A Euler angles and Cartesian coordinates

I have a point A:- Known it's Cartesian coordinates (X,Y,Z) and its Euler angle Aka body rotation (R,P,Y) respectively Roll (rotation around X axis) , Pitch (rotaion around Y axis) and Yaw (rotation ...
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159 views

Conversion of a matrix differential equation for rotation to quaternions

Specifications and Data We have a 3D rotation function $R(t)_{3\times 3}$ and function ${K(t)}_{3\times 3}$, a matrix-valued function that gives skew symmetric matrices as outputs. It has the ...
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2answers
137 views

Do Lipschitz/Hurwitz quaternions satisfy the Ore condition?

The Lipschitz quaternions $L$ are the quaternions with integer coefficients and the Hurwitz quaternions $H$ are the quaternions with coefficients from $\Bbb Z\cup(\Bbb Z+1/2).$ A ring satisfies the ...
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0answers
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Quaternions: Rotation Matrix Derivative

Given Data and Specifications in Question If $q(t)$ represents the position vector as result of rotation with an angular velocity $\omega(t)$ in quaternions, then you can make the relationship ...
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1answer
68 views

Multiplication of Rotation Matrices in quaternion

Given Data and specifications NB : * means multiplication Suppose we need to rotate a point $P = \begin{pmatrix} x\\ y\\ z \end{pmatrix}$ with rotation matrix ${Q}_{3\times3}$ then what we do is ...
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2answers
398 views

Unit Quaternion to a Scalar Power

I'm trying to modify a physics engine for efficiency. Currently, as objects move around the world, their orientation (a quaternion) is updated every frame, by multiplying by the rotation (another ...
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1answer
69 views

Exercise 5.8 from Lie Group, Daniel Bump

In the exercise 5.8 Bump has asked to prove that the group $Sp(4)$ over complex numbers, which is usual complex embedding $U(4)\cap Sp(4,\mathbb{C})$, can be described by, $$\left\{\begin{pmatrix} ...
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162 views

Quaternion Calculus

I was reading a note on Quaternion(Link) and I am happened to read a section regarding a solution of quaternion differential equation. I am putting that segment as picture format here for more ...
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1answer
114 views

Rotation Matrix to Quaternion(proper Orientation)

Given Data in the figure In this figure we have a unit vectors $ x,y,z$ as axis. Axis of rotation is $b$ and angle of rotation is $\phi$. $\phi$ is unknown and $b$ is given as $b= \frac{1}{2 ...
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1answer
113 views

Quaternion Equivalence

Assume $R_{3\times3}$ is a rotation matrix. Question Is it true that there exists two quaternions representing this same rotation matrix $R_{3\times3}$ ? Hint : $\theta = \arccos\left( ...
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1answer
42 views

Quaternionic representation

Let $V$ be $G$-representation over quaternions $\mathbb{H}$. How to show that $$ \mathbb{H} \otimes_\mathbb{C} V $$ is canonically isomorphic to $V \oplus V$ as representation over $\mathbb{H}$? In ...
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1answer
78 views

Showing that $\mathbb{H}^{*}$ maps onto $\mathrm{Aut}(\mathbb{H})$

To show that $\mathbb{H}$ maps onto $\mathrm{Aut}(\mathbb{H})$, where $\mathbb{H}$ is the Hamilton quaternions, I thought it'd be pertinent to show first that the subgroup of inner automorphisms of ...
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0answers
37 views

Cross product uniqueness

I have following relationship between vectors $A_1'(t)=\psi(t)\times A_1(t) \tag1$ $A_2'(t)=\psi(t)\times A_2(t) \tag2$ $A_3'(t)=\psi(t)\times A_3(t) \tag3$ Given Data " ' " means derivative ...
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4answers
203 views

Rotating Frame and Angular Velocity

We have an equation $ \frac{dr}{dt}=\Omega \times \bf r \tag 1$ SPECIFICATIONS $\times$ means cross product,$\Omega$ constant angular velocity,${\bf r}$ is the postion vector of an object Given ...
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1answer
98 views

Do all automorphisms of $\mathbb{H}$ preserve the norm of an element?

Do all automorphisms of $\mathbb{H}$-- the Hamilton quaternions-- preserve the norm of an element? I can't seem to answer this question without using the not-so-elementary fact that all automorphisms ...
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194 views

Inverse of a Rotation matrix

If $R $ is a rotation matrix (determinant 1,orthonormal) can we say that $R^{-1}$ is also a rotation matrix? If yes how do we prove it?
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1answer
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Quaternion integration

If the angular velocity is changing continuously, the following holds true $ q(t)=q(0)\exp\left({\int_{0}^{t}\frac{q_\omega(\tau)}{2}\ d\tau}\right) \tag 1$ Specifications and Data $q(t),q(0)$ ...
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6answers
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What is the motivation for quaternions?

I know imaginary numbers solve $x^2 +1=0$, but what is the motivation for quaternions?
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2answers
420 views

Why should I consider the components $j^2$ and $k^2$ to be $=-1$ in the search for quaternions?

I'm reading a paper about Hamilton's discovery of quaternions and it explains why he failed in his 'theory of triplets' where he tried to make a vector with $3$ dimensions, as an analogy to the ...
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1answer
72 views

Finding quaternion that transforms to particular basis

I want to find a quaternion $x \in{\mathbb{H}} $ that transforms (rotates) the $ i,j,k $ basis to a particular basis. In equations: $$ x i x^{-1} = a_1 $$ $$ x j x^{-1} = a_2 $$ $$ x k x^{-1} = a_3 ...
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1answer
89 views

Construction of Hyper-Complex Numbers

How does one construct a hyper-complex number multiplication table? For example: Quarternions: ...
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0answers
58 views

Strong Approximation for adelic quaternionic groups

Let $H$ be a definite quaternion algebra defined over $\mathbb{Q}$, unramified at $p$, and ramified at $\infty$. Denote by $D$ the multiplicative group $H^\times$ divided by its center. In this ...
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1answer
60 views

There is no group whose quotient by the center is isomorphic to the quaternion group [duplicate]

I have to prove that there is no group $G$ such that $G/Z(G)$ is isomorphic to $Q_8$. Anytone can give me an idea to begin? thanks
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3answers
104 views

Are $i,j,k$ commutative?

I am trying to understand quaternions. I read that Hamilton came up with the great equation: A) $i^2 = j^2 = k^2 = ijk = −1$ In this equation I understand that $i,j,k$ are complex numbers. Later ...
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3answers
158 views

Conversion of rotation matrix to quaternion

We use unit length Quaternion to represent rotations. Following is a general rotation matrix obtained ${\begin{bmatrix}m_{00} & m_{01}&m_{02} \\ m_{10} & m_{11}&m_{12}\\ m_{20} & ...
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0answers
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Find path between two attitudes subject to body rate constraints

Here's my problem. I have an initial orientation and angular velocity of a body and a final orientation and velocity occurring at a specified time in the future. I have control over how input ...
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Rotation plane on the sphere (quarternion)

I asked similar question on stackoverflow but still no answers.http://stackoverflow.com/questions/25185329/image-rotation-with-the-gyro-data-math I assume it is more math than programming problem. ...
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2answers
188 views

Derive a quaternion from three axis

My problem originates from some code that I'm writing to parse an obscure file-type in which a geometric entity is defined in it's own 'local space', and a rotation and translation are provided to ...
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2answers
378 views

Solving for a Rotation Matrix Equation $R_1 R_\mathrm{x} R_2 = R_\mathrm{x}$

I would like to solve for an equation $R_1 R_\mathrm{x} R_2 = R_\mathrm{x}$ where $R_1$, $R_2$, and $R_\mathrm{x}$ are 3x3 rotation matrices, $R_1$ and $R_2$ are known, and $R_\mathrm{x}$ is unknown. ...
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1answer
82 views

Tensor products and isomorphic algebras

I found that $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \simeq \mathbb{C} \oplus\mathbb{C}$ and that $ \mathbb{H} \otimes_{ \mathbb{R}} \mathbb{C} \simeq M_2( \mathbb{C})$. Could anybody hint me how ...
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1answer
193 views

Converting quaternions to spherical angles

Consider a situation where a beam is shot at a cube C from an arbitrary position P. The cube detects the angle of incidence relative to its $ x $ axis. The cube can be rotated and moved, and the ...
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0answers
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can quaternions be expressed in terms of tensor products?

QUESTIONS does this arithmetic check out? if so, is there a geometric interpretation? note: my aim was to try to find a very simple but non-trivial example which might help me begin to understand ...
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1answer
151 views

Proving that $\mathbb R^3$ cannot be made into a real division algebra (and that extending complex multiplication would not work)

I am trying to solve the following exercise: Prove that complex multiplication does not extend to a multiplication on $\mathbb R^3$ so as to make $\mathbb R^3$ into a real division algebra. ...
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4answers
194 views

Need help determining the pairs of quaternions that anticommute

I tried to solve another exercise and I would be grateful if someone could tell me if my answer is right. This is the exercise: Characterize the pairs $p,q \in \mathbb H$ such that $pq = -qp$. I ...
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0answers
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Identifying $\mathbb H^n$ with $\mathbb C^{2n}$

Let $X \in M_n(\mathbb H)$ (Hermetian field). It is possible to make $\mathbb H^n$ into a $2n$-dimensional vector space over $\mathbb C$, for example, by embedding $\mathbb C$ into $\mathbb H$ as ...
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1answer
119 views

Commuting quaternions

I tried to solve the following exercise, please could somebody tell me if I did it right?: Prove that non-real elements $x,y \in \mathbb H$ commute if and only if their imaginary parts are ...