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 ,

From these definitions, it follows that

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,

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

giving the identities

for and , and

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

where 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

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