direction cosines


If the non-zero vector  r=xi+yj+zk  of 3 forms the angles α, β and γ with the positive directions of x-axis, y-axis and z-axis, respectively, then the numbers

cosα,cosβ,cosγ

are the direction cosinesMathworldPlanetmath of the vector. Any triple l,m,n of numbers, which are proportional (http://planetmath.org/Variation) to the direction cosines, are direction numbers of the vector.

If  r=x2+y2+z2  is the of r, we see easily that

cosα=xr,cosβ=yr,cosγ=zr.

Conversely, the componentsPlanetmathPlanetmath of the vector on the coordinate axes may be obtained from

x=rcosα,y=rcosβ,z=rcosγ.

We also see that the direction cosines satisfy

cos2α+cos2β+cos2γ=1.
Title direction cosines
Canonical name DirectionCosines
Date of creation 2013-03-22 17:16:32
Last modified on 2013-03-22 17:16:32
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Definition
Classification msc 15A72
Classification msc 51N20
Related topic MutualPositionsOfVectors
Related topic EquationOfPlane
Defines direction numbers