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

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 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 or obtuse angles, i.e., $0 \leq a \leq π$ , $0 \leq b \leq π$ and $0 \leq c \leq π$ , 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. 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

If vectors $u$ and $v$ have direction cosines $(α u, β u, γ u)$ and $(α v, β v, γ v)$ respectively, with an angle $θ$ between them, their units vectors are

${\begin{aligned}\mathbf {\hat {u}} &={\frac {1}{\sqrt {u_{x}^{2}+u_{y}^{2}+u_{z}^{2}}}}\left(u_{x}\mathbf {e} _{x}+u_{y}\mathbf {e} _{y}+u_{z}\mathbf {e} _{z}\right)=\alpha _{u}\mathbf {e} _{x}+\beta _{u}\mathbf {e} _{y}+\gamma _{u}\mathbf {e} _{z}\\\mathbf {\hat {v}} &={\frac {1}{\sqrt {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}}}\left(v_{x}\mathbf {e} _{x}+v_{y}\mathbf {e} _{y}+v_{z}\mathbf {e} _{z}\right)=\alpha _{v}\mathbf {e} _{x}+\beta _{v}\mathbf {e} _{y}+\gamma _{v}\mathbf {e} _{z}.\end{aligned}}$ Taking the dot product of these two unit vectors yield, $\mathbf {{\hat {u}}\cdot {\hat {v}}} =\alpha _{u}\alpha _{v}+\beta _{u}\beta _{v}+\gamma _{u}\gamma _{v}=\cos \theta ,$ 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,

$\theta =\arccos \left(\alpha _{u}\alpha _{v}+\beta _{u}\beta _{v}+\gamma _{u}\gamma _{v}\right).$

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.

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