Let
be the Angle between
and
,
the Angle between
and
, and
the Angle between
and
. Then the direction cosines are equivalent to the
coordinates
of a Unit Vector
,
![\begin{displaymath}
\alpha \equiv \cos a \equiv {{\bf v}\cdot \hat {\bf x}\over \vert{\bf v}\vert}
\end{displaymath}](d2_170.gif) |
(1) |
![\begin{displaymath}
\beta \equiv \cos b \equiv {{\bf v}\cdot \hat {\bf y}\over \vert{\bf v}\vert}
\end{displaymath}](d2_171.gif) |
(2) |
![\begin{displaymath}
\gamma \equiv \cos c \equiv {{\bf v}\cdot \hat {\bf z}\over \vert{\bf v}\vert}.
\end{displaymath}](d2_172.gif) |
(3) |
From these definitions, it follows that
![\begin{displaymath}
\alpha^2+\beta^2+\gamma^2 = 1.
\end{displaymath}](d2_173.gif) |
(4) |
To find the Jacobian when performing integrals over direction cosines, use
The Jacobian is
![\begin{displaymath}
\left\vert\matrix{\partial(\theta, \phi)\over\partial(\alpha...
...er\partial\beta}\cr}\right\vert.
\hrule width 0pt height 4.3pt
\end{displaymath}](d2_180.gif) |
(8) |
Using
so
Direction cosines can also be defined between two sets of Cartesian Coordinates,
![\begin{displaymath}
\alpha_1 \equiv \hat{\bf x}'\cdot\hat{\bf x}
\end{displaymath}](d2_192.gif) |
(13) |
![\begin{displaymath}
\alpha_2 \equiv \hat{\bf x}'\cdot\hat{\bf y}
\end{displaymath}](d2_193.gif) |
(14) |
![\begin{displaymath}
\alpha_3 \equiv \hat{\bf x}'\cdot\hat{\bf z}
\end{displaymath}](d2_194.gif) |
(15) |
![\begin{displaymath}
\beta_1 \equiv \hat{\bf y}'\cdot\hat{\bf x}
\end{displaymath}](d2_195.gif) |
(16) |
![\begin{displaymath}
\beta_2 \equiv \hat{\bf y}'\cdot\hat{\bf y}
\end{displaymath}](d2_196.gif) |
(17) |
![\begin{displaymath}
\beta_3 \equiv \hat{\bf y}'\cdot\hat{\bf z}
\end{displaymath}](d2_197.gif) |
(18) |
![\begin{displaymath}
\gamma_1 \equiv \hat{\bf z}'\cdot\hat{\bf x}
\end{displaymath}](d2_198.gif) |
(19) |
![\begin{displaymath}
\gamma_2 \equiv \hat{\bf z}'\cdot\hat{\bf y}
\end{displaymath}](d2_199.gif) |
(20) |
![\begin{displaymath}
\gamma_3 \equiv \hat{\bf z}'\cdot\hat{\bf z}.
\end{displaymath}](d2_200.gif) |
(21) |
Projections of the unprimed coordinates onto the primed coordinates yield
and
Projections of the primed coordinates onto the unprimed coordinates yield
and
![\begin{displaymath}
x = {\bf r}\cdot\hat {\bf x} = \alpha_1x+\beta_1y+\gamma_1z
\end{displaymath}](d2_222.gif) |
(31) |
![\begin{displaymath}
y = {\bf r}\cdot\hat {\bf y} = \alpha_2x+\beta_2y+\gamma_2z
\end{displaymath}](d2_223.gif) |
(32) |
![\begin{displaymath}
z = {\bf r}\cdot\hat {\bf z} = \alpha_3x+\beta_3y+\gamma_3z.
\end{displaymath}](d2_224.gif) |
(33) |
Using the orthogonality of the coordinate system, it must be true that
![\begin{displaymath}
\hat {\bf x}\cdot\hat {\bf y} = \hat {\bf y}\cdot\hat {\bf z} = \hat {\bf z}\cdot\hat {\bf x} = 0
\end{displaymath}](d2_225.gif) |
(34) |
![\begin{displaymath}
\hat {\bf x}\cdot\hat {\bf x} = \hat {\bf y}\cdot\hat {\bf y} = \hat {\bf z}\cdot\hat {\bf z} = 1,
\end{displaymath}](d2_226.gif) |
(35) |
giving the identities
![\begin{displaymath}
\alpha_l\alpha_m+\beta_l\beta_m+\gamma_l\gamma_m = 0
\end{displaymath}](d2_227.gif) |
(36) |
for
and
, and
![\begin{displaymath}
{\alpha_l}^2+{\beta_l}^2+{\gamma_l}^2 = 1
\end{displaymath}](d2_230.gif) |
(37) |
for
. These two identities may be combined into the single identity
![\begin{displaymath}
\alpha_l\alpha_m+\beta_l\beta_m+\gamma_l\gamma_m = \delta_{lm},
\end{displaymath}](d2_232.gif) |
(38) |
where
is the Kronecker Delta.
© 1996-9 Eric W. Weisstein
1999-05-24