# 3D Rotations

You should have come from here!

Here’s a quickie: What are the eigenvalues of a 2D rotation matrix?

Here’s a problem: For a bunch of rotations performed one after another on a 3D object, find an equivalent single rotation which would give the same result.

Here’s a solution: First of all, multiply all the rotation matrices (obviously!) such that $R_{s}=R_{n}R_{n-1}\cdots R_{2}R_{1}$. A single rotation is characterized by an axis and an angle of rotation. Let’s get the angle first. Find the eigenvalues of $R_{s}$. Two of them will be of the form: $e^{\dot{\iota}\theta}\,\&\,e^{-\dot{\iota}\theta}$. This θ is the required angle. Let’s get the axis now. So what would be the third eigenvalue? Remember $R_{s} \in SO(3)$ which means the third one is 1! Find the eigenvector corresponding to this eigenvalue which is the required axis!

Here’s an example: Let’s do the problem mentioned in Gravitation.

$R_{1}=\begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$;

$R_{2}=\begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ -1 & 0 & 0 \end{pmatrix}$.

$\Rightarrow R_{s}=R_{2}R_{1}=\begin{pmatrix} 0 & 0 & 1 \\ -1 & 0 & 0 \\ 0 & -1 & 0 \end{pmatrix}$.

Relevant eigenvalues of $R_{s}$ are $-\frac{1}{2}\pm\frac{\sqrt{3}}{2}\dot{\iota}$ which give us the angle: $\mathrm{tan}^{-1}\left(-\sqrt{3}\right)=120^{\circ}$. The eigenvector corresponding to 1 is $\frac{1}{\sqrt{3}}(1,-1,1)$ so the axis is one of the diagonals! This solution agrees with the one given in the book.