Direction cosine

General meaning

More generally, direction cosine refers to the cosine of the angle between any two vectors starting at a same point. 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.

Applications

Determining angles between two vectors

Let u and v have direction cosines (α_u, β_u, γ_u) and (α_v, β_v, γ_v), respectively, having an angle θ between them. Their unit vectors are$${\displaystyle \begin{array}{cc}\hat{\mathbf{u}}& =\frac{u_x}{\Vert \mathbf{u}\Vert }{\mathbf{e}}_x+\frac{u_y}{\Vert \mathbf{u}\Vert }{\mathbf{e}}_y+\frac{u_z}{\Vert \mathbf{u}\Vert }{\mathbf{e}}_z={\alpha }_u{\mathbf{e}}_x+{\beta }_u{\mathbf{e}}_y+{\gamma }_u{\mathbf{e}}_z\\ \hat{\mathbf{v}}& =\frac{v_x}{\Vert \mathbf{v}\Vert }{\mathbf{e}}_x+\frac{v_y}{\Vert \mathbf{v}\Vert }{\mathbf{e}}_y+\frac{v_z}{\Vert \mathbf{v}\Vert }{\mathbf{e}}_z={\alpha }_v{\mathbf{e}}_x+{\beta }_v{\mathbf{e}}_y+{\gamma }_v{\mathbf{e}}_z\end{array}}$$respectively.

Taking the scalar product of these two unit vectors yield,$${\displaystyle \hat{\mathbf{u}}\cdot \hat{\mathbf{v}}={\alpha }_u{\alpha }_v+{\beta }_u{\beta }_v+{\gamma }_u{\gamma }_v.}$$The geometric interpretation of the scalar product of these two unit vectors is equivalent to the projection of one vector onto another; linking the two definitions we find the following.

${\displaystyle {\alpha }_u{\alpha }_v+{\beta }_u{\beta }_v+{\gamma }_u{\gamma }_v=\cos \theta }$

There exist two choices for θ (because cosine is even); one is acute, right, obtuse, or straight angle (0 ≤ θ ≤ π), another is the reflex angle (π < θ < 2π), between the vectors. The convention is to choose the former, so we get θ from the scalar product by using the inverse cosine function:$${\displaystyle \theta =\arccos \left({\alpha }_u{\alpha }_v+{\beta }_u{\beta }_v+{\gamma }_u{\gamma }_v\right).}$$