## Quick Reference

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≤*α*≤*π*, 0≤*β*≤*π*, 0≤*γ*≤*π*). Then cos *α*, cos *β* and cos *γ* are the direction cosines of the given direction or of . They are not independent, however, since cos^{2}*α*+cos^{2}*β*+cos^{2}*γ*=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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