PlanetPhysics/Direction Cosines
The Direction Cosines define the orientation of a vector with respect to a coordinate reference frame. Each direction cosine is the cosine of the angle between the vector and its corresponding coordinate axis. Let us first look at a two dimensional example in figure 1: \newline \begin{figure}[!hhp]
\caption{2D - Direction Cosines} \includegraphics[width=\textwidth]{figure1.eps} \end{figure}
The direction cosines of are
The x coordinate is given from simple trigonometry by
where v is the magnitude of the vector . Similarily, the y coordinate is given by
but we can convert this to a cosine through the trigonometric identity that
From figure 1 we see that
which can be subsitituded into 3 to get
Note that is the angle between the y-axis and , so our vector can be represented in this 2D coordinate frame by
Extending this concept to three dimensions is quite easy, from figure 2 we can define with respect t coordinate frame by
in a more compact form with
we get the relation
The directional cosines for figure 2 are
An important property of the direction cosines is that
One important application is to use the direction cosines to define a coordinate system with reference to another. This can be accompished by defining the location of each coordinate axis unit vector with respect to the 'parent'. Once these nine direction cosines are determined (3 for each unit vector), than a transformation matrix exists to carry out coordinate transformations between the child frame and the parent frame.