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 ,
(1)
 
(2)
 
(3)

From these definitions, it follows that
(4)

To find the Jacobian when performing integrals over direction cosines, use
(5)
 
(6)
 
(7)

The Jacobian is
(8)

Using
(9)
 
(10)

(11)
 
(12)

so
(13)
 
(14)
 
(15)
 
(16)

Direction cosines can also be defined between two sets of Cartesian coordinates,
(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

Projections of the unprimed coordinates onto the primed coordinates yield
(26)
 
(27)
 
(28)
 
(29)
 
(30)
 
(31)

and
(32)
 
(33)
 
(34)
 
(35)
 
(36)
 
(37)

Projections of the primed coordinates onto the unprimed coordinates yield
(38)
 
(39)
 
(40)
 
(41)
 
(42)
 
(43)

and
(44)
 
(45)
 
(46)

Using the orthogonality of the coordinate system, it must be true that
(47)

(48)

giving the identities
(49)

for and , and
(50)

for . These two identities may be combined into the single identity
(51)

where is the Kronecker delta.