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Direction Cosine

Let $a$ be the Angle between ${\bf v}$ and ${\bf x}$, $b$ the Angle between ${\bf v}$ and ${\bf y}$, and $c$ the Angle between ${\bf v}$ and ${\bf z}$. Then the direction cosines are equivalent to the $(x,y,z)$ coordinates of a Unit Vector $\hat {\bf v}$,

\alpha \equiv \cos a \equiv {{\bf v}\cdot \hat {\bf x}\over \vert{\bf v}\vert}
\end{displaymath} (1)

\beta \equiv \cos b \equiv {{\bf v}\cdot \hat {\bf y}\over \vert{\bf v}\vert}
\end{displaymath} (2)

\gamma \equiv \cos c \equiv {{\bf v}\cdot \hat {\bf z}\over \vert{\bf v}\vert}.
\end{displaymath} (3)

From these definitions, it follows that
\alpha^2+\beta^2+\gamma^2 = 1.
\end{displaymath} (4)

To find the Jacobian when performing integrals over direction cosines, use
$\displaystyle \theta$ $\textstyle =$ $\displaystyle \sin^{-1}\left({\sqrt{\alpha^2+\beta^2}\,}\right)$ (5)
$\displaystyle \phi$ $\textstyle =$ $\displaystyle \tan^{-1}\left({\beta \over \alpha}\right)$ (6)
$\displaystyle \gamma$ $\textstyle =$ $\displaystyle \sqrt{1-\alpha^2-\beta^2}.$ (7)

The Jacobian is
\left\vert\matrix{\partial(\theta, \phi)\over\partial(\alpha...\partial\beta}\cr}\right\vert.
\hrule width 0pt height 4.3pt
\end{displaymath} (8)

$\displaystyle {d\over dx}(\sin^{-1} x)$ $\textstyle =$ $\displaystyle {1\over\sqrt{1-x^2}}$ (9)
$\displaystyle {d\over dx}(\tan^{-1} x)$ $\textstyle =$ $\displaystyle {1\over 1+x^2},$ (10)

$\displaystyle \left\vert{\partial(\theta, \phi)\over \partial(\alpha, \beta)}\right\vert$ $\textstyle =$ $\displaystyle \left\vert\begin{array}{ccc}{{1\over 2}(\alpha^2+\beta^2)^{-1/2}2...
& {\alpha^{-1}\over 1+{\beta^2\over \alpha^2}}\end{array}\right\vert\nonumber$  
  $\textstyle =$ $\displaystyle {1\over \sqrt{1-\alpha^2-\beta^2}} {(\alpha^2+\beta^2)^{-1/2}\over 1+{\beta^2\over
\alpha^2}} \left({1+{\beta^2\over \alpha^2}}\right)$  
  $\textstyle =$ $\displaystyle {1\over \sqrt{(\alpha^2+\beta^2)(1-\alpha^2-\beta^2)}},$ (11)

$\displaystyle d\Omega$ $\textstyle =$ $\displaystyle \sin\theta\,d\phi\,d\theta = \sqrt{\alpha^2+\beta^2}
...ial(\theta, \phi)\over\partial (\alpha, \beta )}\right\vert \,d\alpha \, d\beta$  
  $\textstyle =$ $\displaystyle {d\alpha\,d\beta\over\sqrt{1-\alpha^2-\beta^2}} = {d\alpha\,d\beta\over\gamma}.$ (12)

Direction cosines can also be defined between two sets of Cartesian Coordinates,
\alpha_1 \equiv \hat{\bf x}'\cdot\hat{\bf x}
\end{displaymath} (13)

\alpha_2 \equiv \hat{\bf x}'\cdot\hat{\bf y}
\end{displaymath} (14)

\alpha_3 \equiv \hat{\bf x}'\cdot\hat{\bf z}
\end{displaymath} (15)

\beta_1 \equiv \hat{\bf y}'\cdot\hat{\bf x}
\end{displaymath} (16)

\beta_2 \equiv \hat{\bf y}'\cdot\hat{\bf y}
\end{displaymath} (17)

\beta_3 \equiv \hat{\bf y}'\cdot\hat{\bf z}
\end{displaymath} (18)

\gamma_1 \equiv \hat{\bf z}'\cdot\hat{\bf x}
\end{displaymath} (19)

\gamma_2 \equiv \hat{\bf z}'\cdot\hat{\bf y}
\end{displaymath} (20)

\gamma_3 \equiv \hat{\bf z}'\cdot\hat{\bf z}.
\end{displaymath} (21)

Projections of the unprimed coordinates onto the primed coordinates yield
$\displaystyle \hat {\bf x}'$ $\textstyle =$ $\displaystyle (\hat {\bf x}'\cdot\hat {\bf x})\hat {\bf x}
+(\hat {\bf x}'\cdot...
...})\hat {\bf z}
= \alpha_1\hat {\bf x}+\alpha_2\hat {\bf y}+\alpha_3\hat {\bf z}$  
$\displaystyle \hat {\bf y}'$ $\textstyle =$ $\displaystyle (\hat {\bf y}'\cdot\hat {\bf x})\hat {\bf x}
+(\hat {\bf y}'\cdot...
...f z})\hat {\bf z}
= \beta_1\hat {\bf x}+\beta_2\hat {\bf y}+\beta_3\hat {\bf z}$  
$\displaystyle \hat {\bf z}'$ $\textstyle =$ $\displaystyle (\hat {\bf z}'\cdot\hat {\bf x})\hat {\bf x}
+(\hat {\bf z}'\cdot...
...)\hat {\bf z}
= \gamma_1\hat {\bf x}+\gamma_2\hat {\bf y}+\gamma_3\hat {\bf z},$  

$\displaystyle x'$ $\textstyle =$ $\displaystyle {\bf r}\cdot\hat {\bf x}' = \alpha_1x+\alpha_2y+\alpha_3z$ (25)
$\displaystyle y'$ $\textstyle =$ $\displaystyle {\bf r}\cdot\hat {\bf y}' = \beta_1x+\beta_2y+\beta_3z$ (26)
$\displaystyle z'$ $\textstyle =$ $\displaystyle {\bf r}\cdot\hat {\bf z}' = \gamma_1x+\gamma_2y+\gamma_3z.$ (27)

Projections of the primed coordinates onto the unprimed coordinates yield
$\displaystyle \hat {\bf x}$ $\textstyle =$ $\displaystyle (\hat {\bf x}\cdot\hat {\bf x}')\hat {\bf x}'
+(\hat {\bf x}\cdot\hat {\bf y}')\hat {\bf y}'
+(\hat {\bf x}\cdot\hat {\bf z}')\hat {\bf z}'$  
  $\textstyle =$ $\displaystyle \alpha_1\hat {\bf x}'+\beta_1\hat {\bf y}'+\gamma_1\hat {\bf z}'$ (28)
$\displaystyle \hat {\bf y}$ $\textstyle =$ $\displaystyle (\hat {\bf y}\cdot\hat {\bf x}')\hat {\bf x}'
+(\hat {\bf y}\cdot\hat {\bf y}')\hat {\bf y}'+(\hat {\bf y}\cdot\hat {\bf z}')\hat {\bf z}'$  
  $\textstyle =$ $\displaystyle \alpha_2\hat {\bf x}'+\beta_2\hat {\bf y}'+\gamma_2\hat {\bf z}'$ (29)
$\displaystyle \hat {\bf z}$ $\textstyle =$ $\displaystyle (\hat {\bf z}\cdot\hat {\bf x}')\hat {\bf x}'
+(\hat {\bf z}\cdot\hat {\bf x}')\hat {\bf y}'+(\hat {\bf z}\cdot\hat {\bf z}')\hat {\bf z}'$  
  $\textstyle =$ $\displaystyle \alpha_3\hat {\bf x}'+\beta_3\hat {\bf y}'+\gamma_3\hat {\bf z}',$ (30)

x = {\bf r}\cdot\hat {\bf x} = \alpha_1x+\beta_1y+\gamma_1z
\end{displaymath} (31)

y = {\bf r}\cdot\hat {\bf y} = \alpha_2x+\beta_2y+\gamma_2z
\end{displaymath} (32)

z = {\bf r}\cdot\hat {\bf z} = \alpha_3x+\beta_3y+\gamma_3z.
\end{displaymath} (33)

Using the orthogonality of the coordinate system, it must be true that
\hat {\bf x}\cdot\hat {\bf y} = \hat {\bf y}\cdot\hat {\bf z} = \hat {\bf z}\cdot\hat {\bf x} = 0
\end{displaymath} (34)

\hat {\bf x}\cdot\hat {\bf x} = \hat {\bf y}\cdot\hat {\bf y} = \hat {\bf z}\cdot\hat {\bf z} = 1,
\end{displaymath} (35)

giving the identities
\alpha_l\alpha_m+\beta_l\beta_m+\gamma_l\gamma_m = 0
\end{displaymath} (36)

for $l, m = 1, 2, 3$ and $ l \not = m$, and
{\alpha_l}^2+{\beta_l}^2+{\gamma_l}^2 = 1
\end{displaymath} (37)

for $l = 1, 2, 3$. These two identities may be combined into the single identity
\alpha_l\alpha_m+\beta_l\beta_m+\gamma_l\gamma_m = \delta_{lm},
\end{displaymath} (38)

where $\delta_{lm}$ is the Kronecker Delta.

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© 1996-9 Eric W. Weisstein