direction cosines

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In a Cartesian coordinate system in 3-dimensional space, a certain direction can be specified as follows. Take a point P such that has the given direction and |OP|=1. Let α, β and γ be the three angles ∠xOP, ∠yOP and ∠zOP, measured in radians (0≤απ, 0≤βπ, 0≤γπ). Then cos α, cos β and cos γ are the direction cosines of the given direction or of . They are not independent, however, since cos2α+cos2β+cos2γ=1. Point P has coordinates (cos α, cos β, cos γ) and, using the standard unit vectors i, j and k along the coordinate axes, the position vector p of P is given by p=(cos α)i+(cos β)j+(cos γ)k. So the direction cosines are the components of p. The direction cosines of the x-axis are 1, 0, 0; of the y-axis, 0, 1, 0; and of the z-axis, 0, 0, 1.

Subjects: Mathematics.

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