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direction cosines (Definition)

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:

Figure: 2D - Direction Cosines
\includegraphics[width=\textwidth]{figure1.eps}

The direction cosines of $\vec {v}$ are

$\displaystyle d_1 = \cos(\theta)$ (1)
$\displaystyle d_2 = \cos(\phi)$ (2)

The x coordinate is given from simple trigonometry by

$\displaystyle x = v \cos(\theta)$ (3)

where v is the magnitude of the vector $\vec v $ . Similarily, the y coordinate is given by

$\displaystyle y = v \sin(\theta)$ (4)

but we can convert this to a cosine through the trigonometric identity that

$\displaystyle cos( 90 - \theta ) = \sin( \theta )$ (5)
From figure 1 we see that
$\displaystyle \phi = 90^o - \theta$ (6)
which can be subsitituded into 3 to get
$\displaystyle y = v \cos(\phi)$ (7)
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
$\displaystyle \vec v = {v \cos(\theta) } \hat{x} + {v \cos(\phi) } \hat{y}$ (8)
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
$\displaystyle \vec v = {v \cos(\alpha)} \hat{x} + {v \cos(\beta)} \hat{y} + {v \cos(\gamma)} \hat{z}$ (9)
in a more compact form with
$\displaystyle v_1 = v \cos(\alpha)$ (10)
$\displaystyle v_2 = v \cos(\beta)$ (11)
$\displaystyle v_3 = v \cos(\gamma)$ (12)
we get the relation
$\displaystyle \vec v = {\vec v_1} \hat{x} + {\vec v_2} \hat{y} + {\vec v_3} \hat{z}$ (13)

The directional cosines for figure 2 are

$\displaystyle d_1 = \cos(\alpha)$ (14)
$\displaystyle d_2 = \cos(\beta)$ (15)
$\displaystyle d_3 = \cos(\gamma)$ (16)

An important property of the direction cosines is that

$\displaystyle {\alpha}^2 + {\beta}^2 + {\gamma}^2 = 1$ (17)

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.



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Cross-references: matrix, unit vector, system, relation, concept, identity, magnitude, reference frame, vector
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This is version 13 of direction cosines, born on 2004-12-28, modified 2006-03-30.
Object id is 26, canonical name is DirectionCosines.
Accessed 12415 times total.

Classification:
Physics Classification: 45.40.-f (Dynamics and kinematics of rigid bodies)
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