$$ \newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\dx}{\text{ dx}}
\newcommand{\rang}{\text{rang}}
\newcommand{\s}{\ \ \ \ \ \ }
\newcommand{\arrows}{\s \Leftrightarrow \s}
\newcommand{\Arrows}{\s \Longleftrightarrow \s}
\newcommand{\arrow}{\s \Rightarrow \s}
\newcommand{\c}{\bcancel}
\newcommand{\v}[2]{
\begin{pmatrix}
#1 \\
#2 \\
\end{pmatrix}
}
\newcommand{\vt}[3]{
\begin{pmatrix}
#1 \\
#2 \\
#3 \\
\end{pmatrix}
}
\newcommand{\stack}[2]{
\substack{
#1 \\
#2
}
}
\newcommand{\atom}[3]{
\substack{
#1 \\
#2
}
\ce{#3}
}
$$
Axis Angle (Euler Vector)
A way of defining a rotation with an axis and an angle. See slides.
Can be given seperately or combined:
$$ \theta, , \hat{K} = \begin{bmatrix} k_x \ k_y \ k_z \end{bmatrix} \s \text{or} \s K = \theta \hat{K} = \begin{bmatrix} \theta k_x \ \theta k_y \ \theta k_z \end{bmatrix} $$
Calculate the equivalent rotation matrix.
$$ ^A_BR(\ ^AK, \theta\ ) = I \cos(\theta) + KK^{-1} (1 - \cos(\theta)) + \hat{K} \sin(\theta) $$