PlanetPhysics/Direction Cosines

The Direction Cosines define the orientation of a [[../Vectors/|vector]] with respect to a coordinate [[../CosmologicalConstant2/|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 $\overrightarrow{v}$ are

$d_1=\cos (\theta )$

$d_2=\cos (\varphi )$

The x coordinate is given from simple trigonometry by

$x=v\cos (\theta )$

where v is the [[../AbsoluteMagnitude/|magnitude]] of the vector $\overrightarrow{v}$ . Similarily, the y coordinate is given by

$y=v\sin (\theta )$

but we can convert this to a cosine through the trigonometric [[../Cod/|identity]] that

$cos(90-\theta )=\sin (\theta )$

From figure 1 we see that

$\varphi =90^o-\theta$

which can be subsitituded into 3 to get

$y=v\cos (\varphi )$

Note that $\varphi$ is the angle between the y-axis and $\overrightarrow{v}$, so our vector $\overrightarrow{v}$ can be represented in this 2D coordinate frame by

$\overrightarrow{v}=v\cos (\theta )\hat{x}+v\cos (\varphi )\hat{y}$

Extending this [[../PreciseIdea/|concept]] to three dimensions is quite easy, from figure 2 we can define $\overrightarrow{v}$ with respect t $\hat{x},\hat{y},\hat{z}$ coordinate frame by

$\overrightarrow{v}=v\cos (\alpha )\hat{x}+v\cos (\beta )\hat{y}+v\cos (\gamma )\hat{z}$

in a more compact form with

$v_1=v\cos (\alpha )$

$v_2=v\cos (\beta )$

$v_3=v\cos (\gamma )$

we get the [[../Bijective/|relation]]

$\overrightarrow{v}={\overrightarrow{v}}_1\hat{x}+{\overrightarrow{v}}_2\hat{y}+{\overrightarrow{v}}_3\hat{z}$

The directional cosines for figure 2 are

$d_1=\cos (\alpha )$

$d_2=\cos (\beta )$

$d_3=\cos (\gamma )$

An important property of the direction cosines is that

${\alpha }^2+{\beta }^2+{\gamma }^2=1$

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