retningcosine

Retningcosine, commonly known as direction cosines, are the cosines of the angles a nonzero vector makes with the coordinate axes in a three-dimensional orthonormal frame. For a vector v = (x, y, z) with magnitude |v| = sqrt(x^2 + y^2 + z^2), the direction cosines are l = cos α = x/|v|, m = cos β = y/|v|, and n = cos γ = z/|v|, where α, β, γ are the angles with the x-, y-, and z-axes, respectively. The triple (l, m, n) describes the direction of v as a unit vector, independent of its length.

These cosines satisfy l^2 + m^2 + n^2 = 1. They are the ratios of the vector's axis projections to its magnitude, so given coordinates (x, y, z), l = x/|v|, m = y/|v|, and n = z/|v|. Conversely, a unit direction vector with cosines (l, m, n) points along v = (l, m, n) scaled by any positive length.

Angles can be recovered by α = arccos(l), β = arccos(m), γ = arccos(n). The cosines also arise from dot products with the unit basis vectors: l = v · i / |v|, m = v · j / |v|, n = v · k / |v|.

Applications include specifying the orientation of a line or velocity vector, transforming between Cartesian components and angular orientation, and constructing direction cosine matrices for coordinate transformations in physics and engineering. In two dimensions, only two cosines exist with l^2 + m^2 = 1; in higher dimensions, there are n direction cosines whose squares sum to 1.