Direction cosine

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In analytic geometry, the direction cosines (or directional cosines) of a vector are the cosines of the angles between the vector and the three coordinate axes. Or equivalently it is the component contributions of the basis to the unit vector.

Three-dimensional Cartesian coordinates[edit]

Vector v in ℝ3.
.
Further information: Cartesian coordinates

If v is a Euclidean vector in three-dimensional Euclidean space, ℝ3,

{\mathbf v}= v_\text{x} \mathbf{e}_\text{x} + v_\text{y} \mathbf{e}_\text{y} + v_\text{z} \mathbf{e}_\text{z}

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

\begin{align}
\alpha &= \cos a = \frac{{\mathbf v} \cdot \mathbf{e}_\text{x} }{ \left | {\mathbf v} \right | } & = \frac{v_\text{x}}{\sqrt{v_\text{x}^2 + v_\text{y}^2 + v_\text{z}^2}} ,\\
\beta  &= \cos b = \frac{{\mathbf v} \cdot \mathbf{e}_\text{y} }{ \left | {\mathbf v} \right | } & = \frac{v_\text{y}}{\sqrt{v_\text{x}^2 + v_\text{y}^2 + v_\text{z}^2}} ,\\
\gamma  &= \cos c = \frac{{\mathbf v} \cdot \mathbf{e}_\text{z} }{ \left | {\mathbf v} \right | } & = \frac{v_\text{z}}{\sqrt{v_\text{x}^2 + v_\text{y}^2 + v_\text{z}^2}}.
\end{align}

It follows that by squaring each equation and adding the results:

 \cos ^2 a + \cos ^2 b + \cos ^2 c = 1\,.

Here, α, β and γ are the direction cosines and the Cartesian coordinates of the unit vector v/|v|, and a, b and c are the direction angles of the vector v.

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

General meaning[edit]

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.

See also[edit]

References[edit]

  • D. C. Kay (1988). Tensor Calculus. Schaum’s Outlines. McGraw Hill. p. 18-19. ISBN 0-07-033484-6. 
  • M. R. Spiegel, S. Lipschutz, D. Spellman (2009). Vector analysis. Schaum’s Outlines (2nd ed.). McGraw Hill. p. 15, 25. ISBN 978-0-07-161545-7. 
  • Tang, K.T. (2006). Mathematical Methods for Engineers and Scientists 2. Springer. p. 13. ISBN 3-540-30268-9.