# Questions tagged [quaternions]

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

150 questions
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### Why are the solutions of polynomial equations so unconstrained over the quaternions?

An $n$th-degree polynomial has at most $n$ distinct zeroes in the complex numbers. But it may have an uncountable set of zeroes in the quaternions. For example, $x^2+1$ has two zeroes in $\mathbb C$,...
8answers
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### Why is the complex number $z=a+bi$ equivalent to the matrix form $\left(\begin{smallmatrix}a &-b\\b&a\end{smallmatrix}\right)$ [duplicate]

Possible Duplicate: Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1 & 0\end{smallmatrix}\right)$ to $i$ On Wikipedia, it says that: Matrix representation ...
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### Why are the only division algebras over the real numbers the real numbers, the complex numbers, and the quaternions?

Why are the only (associative) division algebras over the real numbers the real numbers, the complex numbers, and the quaternions? Here a division algebra is an associative algebra where every ...
2answers
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### 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. I ...
3answers
949 views

### Representing rotations using quaternions

I'm learning Unity and came across a situation where rotations are represented as Quaternions. I've heard that they where used in computer graphics, but never had to use them until now. What I can't ...
7answers
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### Why are There No “Triernions” (3-dimensional analogue of complex numbers / quaternions)? [duplicate]

Since there are complex numbers (2 dimensions) and quaternions (4 dimensions), it follows intuitively that there ought to be something in between for 3 dimensions ("triernions"). Yet no one uses ...
5answers
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### How do you rotate a vector by a unit quaternion?

Given a 3-variable right-handed vector v that is a translation measured in local space and a unit quaternion representing an orientation from local to world space, how do you use the quaternion to ...
4answers
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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 ...
3answers
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### Quaternions vs Axis angle

Whats the use of representing rotation with quaternions compared to normal axis angle representation? I've been trying to learn quaternions and they make enough sense but as far as I can tell ...
3answers
4k views

### Does anyone know any resources for Quaternions for truly understanding them?

I've been studying Quaternions for a week, on my own. I've learned various facts about them but I still don't understand them. My goal is to understand rotation quaternions specifically. I don't want ...
1answer
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3answers
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### Math beyond Quaternions

Quaternions remove the commutative property and octonions eliminate the associative property can we go any higher and eliminate more properties?
1answer
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### What is the valid range for elements of a Quaternion vector to be used for rotation?

I have seen different people saying different things about this so I'm confused. Assuming that each element of the Quaternion vector represents a rotation along some axis, it does not make sense to be ...