Talk:PlanetPhysics/Direction Cosines

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The Direction Cosines define the orientation of a \htmladdnormallink{vector}{http://planetphysics.us/encyclopedia/Vectors.html} with respect to a coordinate \htmladdnormallink{reference frame}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html}. 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{2D - Direction Cosines} \includegraphics[width=\textwidth]{figure1.eps} \end{center} \end{figure}

The direction cosines of $\vec {v}$ are \begin{equation} d_1 = \cos(\theta) \end{equation} \begin{equation} d_2 = \cos(\phi) \end{equation}

The x coordinate is given from simple trigonometry by \begin{equation} x = v \cos(\theta) \end{equation}

where v is the \htmladdnormallink{magnitude}{http://planetphysics.us/encyclopedia/AbsoluteMagnitude.html} 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 \htmladdnormallink{identity}{http://planetphysics.us/encyclopedia/Cod.html} 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 \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} 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 = {v \cos(\alpha)} \hat{x} + {v \cos(\beta)} \hat{y} + {v \cos(\gamma)} \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 \htmladdnormallink{relation}{http://planetphysics.us/encyclopedia/Bijective.html} \begin{equation} \vec v = {\vec v_1} \hat{x} + {\vec v_2} \hat{y} + {\vec v_3} \hat{z} \end{equation}

The directional cosines for figure 2 are \begin{equation} d_1 = \cos(\alpha) \end{equation} \begin{equation} d_2 = \cos(\beta) \end{equation} \begin{equation} d_3 = \cos(\gamma) \end{equation}

An important property of the direction cosines is that \begin{equation} {\alpha}^2 + {\beta}^2 + {\gamma}^2 = 1 \end{equation}

One important application is to use the direction cosines to define a coordinate \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} with reference to another. This can be accompished by defining the location of each coordinate axis \htmladdnormallink{unit vector}{http://planetphysics.us/encyclopedia/PureState.html} with respect to the 'parent'. Once these nine direction cosines are determined (3 for each unit vector), than a transformation \htmladdnormallink{matrix}{http://planetphysics.us/encyclopedia/Matrix.html} exists to carry out coordinate transformations between the child frame and the parent frame.

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