Direction cosine

Short description: Cosines of the angles between a vector and the coordinate axes

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.

Three-dimensional Cartesian coordinates

Vector v in ${\mathbb{ℝ}}^3$

Direction cosines and direction angles for the unit vector $\frac{\mathbf{𝐯}}{|\mathbf{𝐯}|}$

If v is a Euclidean vector in three-dimensional Euclidean space, ${\mathbb{ℝ}}^3,$

$$\mathbf{𝐯}=v_x{\mathbf{𝐞}}_x+v_y{\mathbf{𝐞}}_y+v_z{\mathbf{𝐞}}_z,$$

where e_x, e_y, e_z are the standard basis in Cartesian coordinates notation, then the direction cosines are

$$\begin{array}{rlrl}\alpha & =\cos a=\frac{\mathbf{𝐯}\cdot {\mathbf{𝐞}}_x}{\Vert \mathbf{𝐯}\Vert }& & =\frac{v_x}{\sqrt{v_x^2+v_y^2+v_z^2}},\\ \beta & =\cos b=\frac{\mathbf{𝐯}\cdot {\mathbf{𝐞}}_y}{\Vert \mathbf{𝐯}\Vert }& & =\frac{v_y}{\sqrt{v_x^2+v_y^2+v_z^2}},\\ \gamma & =\cos c=\frac{\mathbf{𝐯}\cdot {\mathbf{𝐞}}_z}{\Vert \mathbf{𝐯}\Vert }& & =\frac{v_z}{\sqrt{v_x^2+v_y^2+v_z^2}}.\end{array}$$

It follows that by squaring each equation and adding the results

$${\cos }^{2}a+{\cos }^{2}b+{\cos }^{2}c={\alpha }^2+{\beta }^2+{\gamma }^2=1.$$

Here α, β, γ are the direction cosines and the Cartesian coordinates of the unit vector $\frac{\mathbf{𝐯}}{|\mathbf{𝐯}|},$ and a, b, c are the direction angles of the vector v.

The direction angles a, b, c are acute, right, obtuse, or straight angles, i.e., 0 ≤ a ≤ π, 0 ≤ b ≤ π and 0 ≤ c ≤ π (There are two opposite angles between two vectors, and the direction angle $x$ between the vectors is the one satisfying 0 ≤ x ≤ π.), and they denote the angles formed between v and the unit basis vectors e_x, e_y, e_z.

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$$\begin{array}{rl}\hat{\mathbf{u}}& =\frac{u_x}{\Vert \mathbf{𝐮}\Vert }{\mathbf{𝐞}}_x+\frac{u_y}{\Vert \mathbf{𝐮}\Vert }{\mathbf{𝐞}}_y+\frac{u_z}{\Vert \mathbf{𝐮}\Vert }{\mathbf{𝐞}}_z={\alpha }_u{\mathbf{𝐞}}_x+{\beta }_u{\mathbf{𝐞}}_y+{\gamma }_u{\mathbf{𝐞}}_z\\ \hat{\mathbf{v}}& =\frac{v_x}{\Vert \mathbf{𝐯}\Vert }{\mathbf{𝐞}}_x+\frac{v_y}{\Vert \mathbf{𝐯}\Vert }{\mathbf{𝐞}}_y+\frac{v_z}{\Vert \mathbf{𝐯}\Vert }{\mathbf{𝐞}}_z={\alpha }_v{\mathbf{𝐞}}_x+{\beta }_v{\mathbf{𝐞}}_y+{\gamma }_v{\mathbf{𝐞}}_z\\ \end{array}$$respectively.

Taking the scalar product of these two unit vectors yield,$$\hat{\mathbf{u}}\mathbf{\cdot }\hat{\mathbf{v}}={\alpha }_u{\alpha }_v+{\beta }_u{\beta }_v+{\gamma }_u{\gamma }_v.$$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.

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

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:$$\theta =\arccos ({\alpha }_u{\alpha }_v+{\beta }_u{\beta }_v+{\gamma }_u{\gamma }_v).$$

See also

• Cartesian tensor
• Euler angles

References

• Kay, D. C. (1988). Tensor Calculus. Schaum’s Outlines. McGraw Hill. pp. 18–19. ISBN 0-07-033484-6. 
• Spiegel, M. R.; Lipschutz, S.; Spellman, D. (2009). Vector analysis. Schaum’s Outlines (2nd ed.). McGraw Hill. pp. 15, 25. ISBN 978-0-07-161545-7. 
• Tyldesley, J. R. (1975). An introduction to tensor analysis for engineers and applied scientists. Longman. p. 5. ISBN 0-582-44355-5. https://books.google.com/books?id=PODXAAAAMAAJ. 
• Tang, K. T. (2006). Mathematical Methods for Engineers and Scientists. 2. Springer. p. 13. ISBN 3-540-30268-9. 
• Weisstein, Eric W.. "Direction Cosine". http://mathworld.wolfram.com/DirectionCosine.html.