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In analytic geometry, the direction cosines (or directional cosines) of a vector are the
cosine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that ...
s 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 mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
in that direction.


Three-dimensional Cartesian coordinates

If is a
Euclidean vector In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Euclidean vectors can be added and scal ...
in
three-dimensional In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (''coordinates'') are required to determine the position (geometry), position of a point (geometry), poi ...
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
, \mathbf v = v_x \mathbf e_x + v_y \mathbf e_y + v_z \mathbf e_z, where are the
standard basis In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors, each of whose components are all zero, except one that equals 1. For exampl ...
in Cartesian notation, then the direction cosines are \begin \alpha &= \cos a = \frac &&= \frac,\\ \beta &= \cos b = \frac &&= \frac,\\ \gamma &= \cos c = \frac &&= \frac. \end It follows that by squaring each equation and adding the results \cos^2 a + \cos^2 b + \cos^2 c = \alpha^ + \beta^ + \gamma^ = 1. Here are the direction cosines and the Cartesian coordinates of the
unit vector In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
\tfrac, and are the direction angles of the vector . The direction angles are acute or obtuse angles, i.e., , and , and they denote the angles formed between and the unit basis vectors .


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 and have direction cosines and respectively, with an angle between them, their units vectors are \begin \mathbf &= \frac\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 &= \frac\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 Taking the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
of these two unit vectors yield, \mathbf = \alpha_u\alpha_v + \beta_u\beta_v + \gamma_u\gamma_v = \cos \theta, where is the angle between the two unit vectors, and is also the angle between and . Since 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).


See also

* Cartesian tensor * Euler angles


References

* * * * * Algebraic geometry Vectors (mathematics and physics) {{Geometry-stub