In analytic geometry, the direction cosines (or directional cosines) of a

vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...

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 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. Vectors can be added to other vectors ...

three-dimensional Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...

, R³, :

\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 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 whose components are all zero, except one that equals 1. For example, in th ...

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 ''α'', ''β'' and ''γ'' 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 (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction ve ...

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 ≤ ''b'' ≤ ''π'' and 0 ≤ ''c'' ≤ ''π'', and they denote the angles formed between v and the unit basis vectors, e_''x'', e_''y'' and e_''z''.

General meaning

More generally, direction cosine refers to the cosine of the angle between any two

s. They are useful for forming direction cosine matrices that express one set of

orthonormal In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of un ...

basis vectors in terms of another set, or for expressing a known

in a different basis.

References

* * * * *{{MathWorld, title=Direction Cosine, urlname=DirectionCosine, url=http://mathworld.wolfram.com/DirectionCosine.html Algebraic geometry Vectors (mathematics and physics)

Three-dimensional Cartesian coordinates

General meaning

See also

References