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.

Three-dimensional Cartesian coordinates

Vector v in ⁠$\mathbb {R} ^{3}$⁠

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

Further information: Cartesian coordinates

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

$${\begin{alignedat}{2}\alpha &{}=\cos a={\frac {\mathbf {v} \cdot \mathbf {e} _{x}}{\Vert \mathbf {v} \Vert }}&&{}={\frac {v_{x}}{\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}},\\\beta &{}=\cos b={\frac {\mathbf {v} \cdot \mathbf {e} _{y}}{\Vert \mathbf {v} \Vert }}&&{}={\frac {v_{y}}{\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}},\\\gamma &{}=\cos c={\frac {\mathbf {v} \cdot \mathbf {e} _{z}}{\Vert \mathbf {v} \Vert }}&&{}={\frac {v_{z}}{\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}}.\end{alignedat}}$$

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 ${\tfrac {\mathbf {v} }{|\mathbf {v} |}},$ 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$${\begin{aligned}\mathbf {\hat {u}} &={\frac {u_{x}}{\rVert \mathbf {u} \lVert }}\mathbf {e} _{x}+{\frac {u_{y}}{\rVert \mathbf {u} \lVert }}\mathbf {e} _{y}+{\frac {u_{z}}{\rVert \mathbf {u} \lVert }}\mathbf {e} _{z}=\alpha _{u}\mathbf {e} _{x}+\beta _{u}\mathbf {e} _{y}+\gamma _{u}\mathbf {e} _{z}\\\mathbf {\hat {v}} &={\frac {v_{x}}{\rVert \mathbf {v} \lVert }}\mathbf {e} _{x}+{\frac {v_{y}}{\rVert \mathbf {v} \lVert }}\mathbf {e} _{y}+{\frac {v_{z}}{\rVert \mathbf {v} \lVert }}\mathbf {e} _{z}=\alpha _{v}\mathbf {e} _{x}+\beta _{v}\mathbf {e} _{y}+\gamma _{v}\mathbf {e} _{z}\\\end{aligned}}$$respectively.

Taking the scalar product of these two unit vectors yield,$$\mathbf {{\hat {u}}\cdot {\hat {v}}} =\alpha _{u}\alpha _{v}+\beta _{u}\beta _{v}+\gamma _{u}\gamma _{v}.$$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:$$\theta =\arccos \left(\alpha _{u}\alpha _{v}+\beta _{u}\beta _{v}+\gamma _{u}\gamma _{v}\right).$$

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.
• Tang, K. T. (2006). Mathematical Methods for Engineers and Scientists. Vol. 2. Springer. p. 13. ISBN 3-540-30268-9.
• Weisstein, Eric W. "Direction Cosine". MathWorld.