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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^^,

alpha=cosa=(v·x^^)/(|v|)
(1)
beta=cosb=(v·y^^)/(|v|)
(2)
gamma=cosc=(v·z^^)/(|v|).
(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)(sqrt(alpha^2+beta^2))
(5)
phi=tan^(-1)(beta/alpha)
(6)
gamma=sqrt(1-alpha^2-beta^2).
(7)

The Jacobian is

 |(partial(theta,phi))/(partial(alpha,beta))|=|(partialtheta)/(partialalpha) (partialtheta)/(partialbeta); (partialphi)/(partialalpha) (partialphi)/(partialbeta)|.
(8)

Using

d/(dx)(sin^(-1)x)=1/(sqrt(1-x^2))
(9)
d/(dx)(tan^(-1)x)=1/(1+x^2),
(10)
|(partial(theta,phi))/(partial(alpha,beta))|=|(1/2(alpha^2+beta^2)^(-1/2)2alpha)/(sqrt(1-alpha^2-beta^2)) (1/2(alpha^2+beta^2)^(-1/2)2beta)/(sqrt(1-alpha^2-beta^2)); (-alpha^(-2)beta)/(1+(beta^2)/(alpha^2)) (alpha^(-1))/(1+(beta^2)/(alpha^2))|
(11)
=1/(sqrt((alpha^2+beta^2)(1-alpha^2-beta^2))),
(12)

so

dOmega=sinthetadphidtheta
(13)
=sqrt(alpha^2+beta^2)|(partial(theta,phi))/(partial(alpha,beta))|dalphadbeta
(14)
=(dalphadbeta)/(sqrt(1-alpha^2-beta^2))
(15)
=(dalphadbeta)/gamma.
(16)

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

x^^^'=(x^^^'·x^^)x^^+(x^^^'·y^^)y^^+(x^^^'·z^^)z^^
(26)
=alpha_1x^^+alpha_2y^^+alpha_3z^^
(27)
y^^^'=(y^^^'·x^^)x^^+(y^^^'·y^^)y^^+(y^^^'·z^^)z^^
(28)
=beta_1x^^+beta_2y^^+beta_3z^^
(29)
z^^^'=(z^^^'·x^^)x^^+(z^^^'·y^^)y^^+(z^^^'·z^^)z^^
(30)
=gamma_1x^^+gamma_2y^^+gamma_3z^^,
(31)

and

x^'=r·x^^^'
(32)
=alpha_1x+alpha_2y+alpha_3z
(33)
y^'=r·y^^^'
(34)
=beta_1x+beta_2y+beta_3z
(35)
z^'=r·z^^^'
(36)
=gamma_1x+gamma_2y+gamma_3z.
(37)

Projections of the primed coordinates onto the unprimed coordinates yield

x^^=(x^^·x^^^')x^^^'+(x^^·y^^^')y^^^'+(x^^·z^^^')z^^^'
(38)
=alpha_1x^^^'+beta_1y^^^'+gamma_1z^^^'
(39)
y^^=(y^^·x^^^')x^^^'+(y^^·y^^^')y^^^'+(y^^·z^^^')z^^^'
(40)
=alpha_2x^^^'+beta_2y^^^'+gamma_2z^^^'
(41)
z^^=(z^^·x^^^')x^^^'+(z^^·x^^^')y^^^'+(z^^·z^^^')z^^^'
(42)
=alpha_3x^^^'+beta_3y^^^'+gamma_3z^^^',
(43)

and

x=r·x^^=alpha_1x^'+beta_1y^'+gamma_1z^'
(44)
y=r·y^^=alpha_2x^'+beta_2y^'+gamma_2z^'
(45)
z=r·z^^=alpha_3x^'+beta_3y^'+gamma_3z^'.
(46)

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 Web Resource. https://mathworld.wolfram.com/DirectionCosine.html

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