# Find rotation matrix to transform from global to local coordinates

How do I find the rotation matrix to transform from the global coordinate frame $(x,y,z)$ to a rotated local coordinate frame $(x',y',z')$, given I have the orthonormal basis vectors of the local coordinate frame?

For example; I have a coordinate frame whose basis vectors are $$x' = (0.5774\;;\; 0.5774\;;\;0.5774)$$$$y' = (-0.8165\;;\; 0.4082\;;\;0.4082)$$$$z' = (0\;;\; -0.7071\;;\;0.7071)$$

How do find $R$ such that $$R\begin{pmatrix} 1 \\0\\0\end{pmatrix} = \begin{pmatrix} 0.5774 \\0.5774\\0.5774\end{pmatrix}$$

• Welcome to Math.SE! In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. – emma Jul 13 '17 at 17:05
• Hints: The columns of a transformation matrix are the images of the basis vectors, and the inverse of a rotation matrix is its transpose. – amd Jul 13 '17 at 20:17