Direction cosine

In analytic geometry, the direction cosines (or directional cosines) of a vector are the cosines of the angles between the vector and the three positive coordinate axes.

Equivalently, they are the contributions of each component of the basis to a unit vector in that direction.

If v is a Euclidean vector in three-dimensional Euclidean space, ⁠

where ex, ey, ez are the standard basis in Cartesian notation, then the direction cosines are

= cos ⁡ a =

= cos ⁡ b =

γ

= cos ⁡ c =

It follows that by squaring each equation and adding the results

cos

cos

cos

α

γ

Here α, β, γ are the direction cosines and the Cartesian coordinates of the unit vector

and a, b, c are the direction angles of the vector v. The direction angles a, b, c are acute or obtuse angles, i.e., 0 ≤ a ≤ π, 0 ≤ b ≤ π and 0 ≤ c ≤ π, and they denote the angles formed between v and the unit basis vectors ex, ey, ez.

More generally, direction cosine refers to the cosine of the angle between any two vectors.

They are useful for forming direction cosine matrices that express one set of orthonormal basis vectors in terms of another set, or for expressing a known vector in a different basis.

Simply put, direction cosines provide an easy method of representing the direction of a vector in a Cartesian coordinate system.

If vectors u and v have direction cosines (αu, βu, γu) and (αv, βv, γv) respectively, with an angle θ between them, their units vectors are

α

γ

α

Taking the dot product of these two unit vectors yield,

= cos ⁡ θ ,

where θ is the angle between the two unit vectors, and is also the angle between u and v. Since θ is a geometric angle, and is never negative.

Therefore only the positive value of the dot product is taken, yielding us the final result,

θ = arccos ⁡

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