# direction cosines

If the non-zero vector $\overrightarrow{r}=x\overrightarrow{i}+y\overrightarrow{j}+z\overrightarrow{k}$ of ${\mathbb{R}}^{3}$ forms the angles $\alpha $, $\beta $ and $\gamma $ with the positive directions of $x$-axis, $y$-axis and $z$-axis, respectively, then the numbers

$$\mathrm{cos}\alpha ,\mathrm{cos}\beta ,\mathrm{cos}\gamma $$ |

are the direction cosines^{} 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 $r=\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}$ is the of $\overrightarrow{r}$, we see easily that

$$\mathrm{cos}\alpha =\frac{x}{r},\mathrm{cos}\beta =\frac{y}{r},\mathrm{cos}\gamma =\frac{z}{r}.$$ |

Conversely, the components^{} of the vector on the coordinate axes may be obtained from

$$x=r\mathrm{cos}\alpha ,y=r\mathrm{cos}\beta ,z=r\mathrm{cos}\gamma .$$ |

We also see that the direction cosines satisfy

$${\mathrm{cos}}^{2}\alpha +{\mathrm{cos}}^{2}\beta +{\mathrm{cos}}^{2}\gamma =1.$$ |

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 |