Given a unit vector $\vec{n}$, find the matrix of rotation about $\vec{n}$.

Question

In $\mathbb{R}^3$ suppose I have an arbitrary vector $\vec{n}$. I want to find the rotation matrix about $\vec{n}$ through an angle $\theta$.

I can develop the rotation matrix in two dimensions and that makes it clear to me that the rotation matrices about the basis vectors are

$$R_x=\begin{pmatrix}1 & 0 & 0 \\0 & \cos\theta &-\sin\theta\\ 0 & \sin\theta & \cos\theta\end{pmatrix}$$

and similarly for $R_y$ and $R_z$. I've also looked at this answer: Vector rotation but I'm trying to follow the steps to deriving it, not just looking for the answer. Also, I know nothing about tensors.

I looked at this answer: Getting a transformation matrix from a normal vector I believe I understand why we would want a rotation that would take the plane normal to the normal, and align it with the $x$-$y$ plane, since having done that we might be able to then follow with a rotation about the $z$ axis to get the resulting vectors aligned with $x$ and $y$ axes. The composition of these rotations would map the new bases to the old, and the inverse would map the old to the new.

So the task is to now rotate $\vec{n}$ onto $\vec{k}$, and I can see (somewhat) why we would want a vector $\vec{u}$ in the $x$-$y$ plane which is normal to this rotation, and how to get it by solving $\vec{n}\cdot \vec{u} = 0$ with $\vec{u} = a\vec{i}+b\vec{j}$.

But once having $\vec{u}$, I'm not sure how I would find the matrix of the rotation about $\vec{u}$ which takes $\vec{n}$ onto $\vec{k}$, and the answer given there points to a Wikipedia article to do this. But again I don't just want the answer, I want to understand the derivation.

Besides that I'm somewhat confused by the methodology in this sense: Don't we have a situation that is not very much better than the situation we started with? Right now I have a vector $\vec{u}$ and want to rotate $\vec{n}$ about it ... Well isn't that the same as what we're trying to do generally? Is anything gained by the fact that $\vec{u}$ is in the $x$-$y$ plane which makes that easier than just doing the same task with $\vec{n}$ directly?

And I would really prefer a derivation that went more like the way that the 2-D rotation proceeded, since I understood that clearly. It would be nice to consider vector $\vec{n}$ and the rotation of $\vec{i}$ about it, write those coordinates into the first column of the answer, and then do likewise for $\vec{j}$ and $\vec{k}$. Of course, perhaps we don't do it that way because it's very hard, and hence why I can't see how to do it that way on my own.

Answer 1

I am going to derive the rotation around a unit vector very slowly.

Given a unit vector $N$ and a vector $X$, the task is to rotate $X$ around $N$.

First suppose we are in the simple case in which the vector $X$ lies in the plane perpendicular to $N$, so:

$X \cdot N = 0$

To rotate $X$ on the plane we first need to find a orthonormal basis on that plane, made of two vectors $U_1$ and $U_2$.

$U1 \cdot U2 = 0$

$U1 \cdot U1 = U2 \cdot U2 = 1$

Since $X$ is already on the rotation plane we can use it as one of our basis vectors.

$U_1 = X / \|X\|$

$U_2 = N \times U_1$

Then we can form a $3 \times 3$ change-of-basis matrix $M$ that has as columns the vectors $U_1$, $U_2$ and $N$.

$M = [U_1 U_2 N]$

The matrix $M$ has the property of sending the cartesian basis vectors $E_1 = [1,0,0]$, $E_2 = [0,1,0]$ and $E_3 = [0,0,1]$ to our rotation plane.

You can check that:

$M E_1 = U_1$

$M E_2 = U_2$

$M E_3 = N$

Since $M$ is an orthonormal matrix its transpose is equal to the inverse $M^T = M^{-1}$ so:

$M^T U_1 = E_1$

$M^T U_2 = E_2$

$M^T N = E_3$

Now, let $R_3$ be a $3\times3$ rotation matrix around the cartesian vector $E_3$ (i.e., a rotation in the plane $E_1 E_2$).

$R = M R_3 M^T$

$R$ is a rotation in the plane $U_1$ and $U_2$ around $N$. You can check that:

$R X = X'$

$(M R_3 M^T) X = X'$

Since $M^T X = \|X\| E_1$ we have:

$M R_3 (M^T X) = M R_3 (\|X\| E_1)$

Lets call $E_1'$ to the rotation of $E_1$ by matrix $R_3$.

$\|X\| M (R_3 E_1) = \|X\| M E_1' $

$M (\|X\| E_1') = X'$

And we get back:

$R X = X'$

Now lets remove the assumption that $X$ is in the rotation plane. We still need to find two vectors $U_1$ and $U_2$ in the rotation plane, so we project $X$ to the rotation plane to find $U_1$ and then we get $U_2$ as the cross product of $N$ and $U_1$.

$U1 = (X - (X \cdot N) N) / \|X - (X \cdot N) N\|$

$U_2 = N \times U_1$

$X$ is a linear combination of vectors $U_1$ and $N$. So applying the matrix $M^T = [U_1 U_2 N]^T$ to $X$ gives:

$M^T X = a E_1 + b E_3$

Where:

$a = \|X - (X \cdot N) N\| $

$b = (X \cdot N)$

Since $E_3$ is not changed by the rotation $R_3$. The rotation $R = M R_3 M^T$ around $N$ is going to change only the component of $X$ projected on the rotation plane. That does the "trick" of rotation around an axis.

Up to now we have used the vector $X$ for creating the vector $U_1$ in the rotation plane. But what happen if we only have the vector $N$ and the rotation angle? In that case we can choose any cartesian vector $E1$, $E2$ or $E3$ provided that they are linearly independent with $N$.

$U_1 = N \times E_1$ or $U_1 = N \times E_2$ or $U_1 = N \times E_3$

Of course there are better ways to choose the basis vectors, see the Rodrigues Formula derivation in wikipedia and perhaps you can understand now how it works.

https://en.m.wikipedia.org/wiki/Rodrigues%27_rotation_formula#Derivation