Direction Cosine
Let a be the angle between v and x, b the angle between v and y, and c the angle between v and z. Then the direction cosines are equivalent to the (x,y,z) coordinates of a unit vector v^^,
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
The Jacobian is
Using
so
Direction cosines can also be defined between two sets of Cartesian coordinates,
| alpha_1=x^^^'·x^^ |
(17)
|
| alpha_2=x^^^'·y^^ |
(18)
|
| alpha_3=x^^^'·z^^ |
(19)
|
| beta_1=y^^^'·x^^ |
(20)
|
| beta_2=y^^^'·y^^ |
(21)
|
| beta_3=y^^^'·z^^ |
(22)
|
| gamma_1=z^^^'·x^^ |
(23)
|
| gamma_2=z^^^'·y^^ |
(24)
|
| gamma_3=z^^^'·z^^. |
(25)
|
Projections of the unprimed coordinates onto the primed coordinates yield
and
Projections of the primed coordinates onto the unprimed coordinates yield
and
Using the orthogonality of the coordinate system, it must be true that
| x^^·y^^=y^^·z^^=z^^·x^^=0 |
(47)
|
| x^^·x^^=y^^·y^^=z^^·z^^=1, |
(48)
|
giving the identities
| alpha_lalpha_m+beta_lbeta_m+gamma_lgamma_m=0 |
(49)
|
for l,m=1,2,3 and l!=m, and
| alpha_l^2+beta_l^2+gamma_l^2=1 |
(50)
|
for l=1,2,3. These two identities may be combined into the single identity
| alpha_lalpha_m+beta_lbeta_m+gamma_lgamma_m=delta_(lm), |
(51)
|
where delta_(lm) is the Kronecker delta.
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Cite this as:
Weisstein, Eric W. "Direction Cosine." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DirectionCosine.html