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 $\hat {\bf v}$,

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

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

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

From these definitions, it follows that

$$\alpha^2+\beta^2+\gamma^2 = 1.$$ (4)

To find the Jacobian when performing integrals over direction cosines, use

$$\theta = \sin^{-1}\left({\sqrt{\alpha^2+\beta^2}\,}\right)$$ (5)

$$\phi = \tan^{-1}\left({\beta \over \alpha}\right)$$ (6)

$$\gamma = \sqrt{1-\alpha^2-\beta^2}.$$ (7)

The Jacobian is

$$\left\vert\matrix{\partial(\theta, \phi)\over\partial(\alpha... ...er\partial\beta}\cr}\right\vert. \hrule width 0pt height 4.3pt$$ (8)

Using

$${d\over dx}(\sin^{-1} x) = {1\over\sqrt{1-x^2}}$$ (9)

$${d\over dx}(\tan^{-1} x) = {1\over 1+x^2},$$ (10)

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

$$= {1\over \sqrt{1-\alpha^2-\beta^2}} {(\alpha^2+\beta^2)^{-1/2}\over 1+{\beta^2\over \alpha^2}} \left({1+{\beta^2\over \alpha^2}}\right)$$

$$= {1\over \sqrt{(\alpha^2+\beta^2)(1-\alpha^2-\beta^2)}},$$ (11)

so

$$d\Omega = \sin\theta\,d\phi\,d\theta = \sqrt{\alpha^2+\beta^2} \left\vert{\... ...ial(\theta, \phi)\over\partial (\alpha, \beta )}\right\vert \,d\alpha \, d\beta$$

$$= {d\alpha\,d\beta\over\sqrt{1-\alpha^2-\beta^2}} = {d\alpha\,d\beta\over\gamma}.$$ (12)

Direction cosines can also be defined between two sets of Cartesian Coordinates,

$$\alpha_1 \equiv \hat{\bf x}'\cdot\hat{\bf x}$$ (13)

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

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

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

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

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

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

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

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

Projections of the unprimed coordinates onto the primed coordinates yield

$$\hat {\bf x}' = (\hat {\bf x}'\cdot\hat {\bf x})\hat {\bf x} +(\hat {\bf x}'\cdot... ...})\hat {\bf z} = \alpha_1\hat {\bf x}+\alpha_2\hat {\bf y}+\alpha_3\hat {\bf z}$$ (22)

$$\hat {\bf y}' = (\hat {\bf y}'\cdot\hat {\bf x})\hat {\bf x} +(\hat {\bf y}'\cdot... ...f z})\hat {\bf z} = \beta_1\hat {\bf x}+\beta_2\hat {\bf y}+\beta_3\hat {\bf z}$$ (23)

$$\hat {\bf z}' = (\hat {\bf z}'\cdot\hat {\bf x})\hat {\bf x} +(\hat {\bf z}'\cdot... ...)\hat {\bf z} = \gamma_1\hat {\bf x}+\gamma_2\hat {\bf y}+\gamma_3\hat {\bf z},$$ (24)

and

$$x' = {\bf r}\cdot\hat {\bf x}' = \alpha_1x+\alpha_2y+\alpha_3z$$ (25)

$$y' = {\bf r}\cdot\hat {\bf y}' = \beta_1x+\beta_2y+\beta_3z$$ (26)

$$z' = {\bf r}\cdot\hat {\bf z}' = \gamma_1x+\gamma_2y+\gamma_3z.$$ (27)

Projections of the primed coordinates onto the unprimed coordinates yield

$$\hat {\bf x} = (\hat {\bf x}\cdot\hat {\bf x}')\hat {\bf x}' +(\hat {\bf x}\cdot\hat {\bf y}')\hat {\bf y}' +(\hat {\bf x}\cdot\hat {\bf z}')\hat {\bf z}'$$

$$= \alpha_1\hat {\bf x}'+\beta_1\hat {\bf y}'+\gamma_1\hat {\bf z}'$$ (28)

$$\hat {\bf y} = (\hat {\bf y}\cdot\hat {\bf x}')\hat {\bf x}' +(\hat {\bf y}\cdot\hat {\bf y}')\hat {\bf y}'+(\hat {\bf y}\cdot\hat {\bf z}')\hat {\bf z}'$$

$$= \alpha_2\hat {\bf x}'+\beta_2\hat {\bf y}'+\gamma_2\hat {\bf z}'$$ (29)

$$\hat {\bf z} = (\hat {\bf z}\cdot\hat {\bf x}')\hat {\bf x}' +(\hat {\bf z}\cdot\hat {\bf x}')\hat {\bf y}'+(\hat {\bf z}\cdot\hat {\bf z}')\hat {\bf z}'$$

$$= \alpha_3\hat {\bf x}'+\beta_3\hat {\bf y}'+\gamma_3\hat {\bf z}',$$ (30)

and

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

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

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

Using the orthogonality of the coordinate system, it must be true that

$$\hat {\bf x}\cdot\hat {\bf y} = \hat {\bf y}\cdot\hat {\bf z} = \hat {\bf z}\cdot\hat {\bf x} = 0$$ (34)

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

giving the identities

$$\alpha_l\alpha_m+\beta_l\beta_m+\gamma_l\gamma_m = 0$$ (36)

for $l, m = 1, 2, 3$ and $l \not = m$, and

$${\alpha_l}^2+{\beta_l}^2+{\gamma_l}^2 = 1$$ (37)

for $l = 1, 2, 3$. These two identities may be combined into the single identity

$$\alpha_l\alpha_m+\beta_l\beta_m+\gamma_l\gamma_m = \delta_{lm},$$ (38)

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