DirectionCosineMatrix 2005-08-25 19:56:33 2005-08-26 15:49:37 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) \usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here A direction cosine matrix (DCM) is a transformation matrix that transforms one coordinate reference frame to another. If we extend the concept of how the three dimensional direction cosines locate a vector, then the DCM locates three unit vectors that describe a coordinate reference frame. Using the notation in equation 1, we need to find the matrix elements that correspond to the correct transformation matrix. \begin{equation} DCM = \left[ \begin{array}{ccc} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{array} \right] \end{equation} The first unit vector of the second coordinate frame can be located in the first frame by normal vector notation. See figure 1 for relationship. \begin{center} $ \hat{y}_1 = A_{11} \hat{x}_1 + A_{12} \hat{x}_2 + A_{13} \hat{x}_3 $ \end{center} \medskip \begin{figure} \includegraphics[scale=0.78]{DCM.eps} \end{figure} \medskip Similarily, the other two unit vectors can be described by \begin{center} $$ \hat{y}_2 = A_{21} \hat{x}_1 + A_{22} \hat{x}_2 + A_{23} \hat{x}_3 $$ $$ \hat{y}_3 = A_{31} \hat{x}_1 + A_{32} \hat{x}_2 + A_{33} \hat{x}_3 $$ \end{center}