Quaternion rotation matrix

Question

I'm a last year student at a high school in Denmark, and I'm trying to understand how quaternions rotations can be described through a rotation matrix. As of now, I quite don't understand how I go from the quaternion: $q = a + bi + cj + dk$, to a quaternion written on matrix form.

I know the quaternion can be written as a structure consisting of the real part and a vector $(a, v)$, where the vector represents the imaginary part of the quaternion, but I don't quite understand how I from the vector part of the can describe a rotation with a rotation matrix.

Answer 1

It's very simple. The key is to only pay attention to a basis.

The $x,y,z$ unit vectors $e_1, e_2, e_3$ of $\mathbb R^3$ are represented by $i,j,k$, respectively. You have a quaternion $q$ such that $x\mapsto qxq^{-1}$ is your rotation.

So in particular, pay attention to $v_1=qiq^{-1}$, $v_2=qjq^{-1}$ and $v_3=qkq^{-1}$. Each $v_i$ is a quaternion, a linear combination of $i,j,k$, and now extract their coefficients of $v_i$ into a $3\times 1$-vector $\vec{v_i}$.

Let $T$ be the matrix whose first column is $\vec{v_1}$, second column is $\vec{v_2}$ and third column is $\vec{v_3}$. Are you surprised to learn we're already done?

Look at what has been produced: $x\mapsto Tx$ ($x$ a column vector), and $x\mapsto qxq^{-1}$ ($x$ a quaternion with real part zero) are both $\mathbb R$-linear transformations, the first one on $\mathbb R^3$ and the second on the quaternions with real part zero. When you compute the image of a vector using either method, you always get results that agree (remember, you only have to know that they agree on the basis vectors!)