direction cosines
DirectionCosines
2004-12-28 19:58:41
2004-12-28 22:48:38
Definition
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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]
\begin{center}
\caption{figure 1: 2D - Direction Cosines}
\includegraphics[width=\textwidth]{figure1.eps}
\end{center}
\end{figure}
The x coordinate is given from simple trigonometry by
\begin{equation}
x = v cos(\theta)
\end{equation}
where v is the magnitude of the vector $ \vec v $ . Similarily, the y coordinate is given by
\begin{equation}
y = v sin(\theta)
\end{equation}
but we can convert this to a cosine through the trigonometric relation that
\begin{equation}
cos( 90 - \theta ) = sin( \theta )
\end{equation}
From figure 1 we see that
\begin{equation}
\phi = 90^o - \theta
\end{equation}
which can be subsitituded into 3 to get
\begin{equation}
y = v cos(\phi)
\end{equation}
Note that $\phi$ is the angle between the y-axis and $\vec v$, so our vector $\vec v$ can be represented in this 2D coordinate frame by
\begin{equation}
\vec v = {v cos(\theta) } \hat{x} + {v cos(\phi) } \hat{y}
\end{equation}
Extending this concept to three dimensions is quite easy, from figure 2 we can define $\vec v$ with respect t $\hat{x}, \hat{y}, \hat{z}$ coordinate frame by
\begin{equation}
\vec v = {\vec v_1} \hat{x} + {\vec v_2} \hat{y} + {\vec v_3} \hat{z}
\end{equation}
in a more compact form with
\begin{equation}
v_1 = v cos(\alpha)
\end{equation}
\begin{equation}
v_2 = v cos(\beta)
\end{equation}
\begin{equation}
v_3 = v cos(\gamma)
\end{equation}
we get the relation
\begin{equation}
\vec v = {v cos(\alpha)} \hat{x} + {v cos(\beta)} \hat{y} + {v cos(\gamma)} \hat{z}
\end{equation}
\begin{figure}[!hhp]
\begin{center}
\caption{figure 1: 3D - Direction Cosines}
\includegraphics[width=\textwidth]{figure2.eps}
\end{center}
\end{figure}
An important property of the direction cosines is that
\begin{equation}
{\alpha}^2 + {\beta}^2 + {\gamma}^2 = 1
\end{equation}