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
The Jacobian is

(8) 
Using
so
Direction cosines can also be defined between two sets of Cartesian Coordinates,

(13) 

(14) 

(15) 

(16) 

(17) 

(18) 

(19) 

(20) 

(21) 
Projections of the unprimed coordinates onto the primed coordinates yield
and
Projections of the primed coordinates onto the unprimed coordinates yield
and

(31) 

(32) 

(33) 
Using the orthogonality of the coordinate system, it must be true that

(34) 

(35) 
giving the identities

(36) 
for
and , and

(37) 
for . These two identities may be combined into the single identity

(38) 
where is the Kronecker Delta.
© 19969 Eric W. Weisstein
19990524