# angle between line and plane

The angle between a line $l$ and a plane $\tau $ is defined as the least possible angle $\omega $ between $l$ and a line contained by $\tau $.

It is apparent that $\omega $ satisfies always $0\leqq \omega \leqq {90}^{\circ}$.

Let the plane $\tau $ be given by the equation (http://planetmath.org/EquationOfPlane) $Ax+By+Cz+D=0$, i.e. its normal vector^{} has the components^{} $A,B,C$. Let a direction vector of the line $l$ have the components $a,b,c$. Then the angle $\omega $ between $l$ and $\tau $ is obtained from the equation

$$\mathrm{sin}\omega =\frac{|Aa+Bb+Cc|}{\sqrt{{A}^{2}+{B}^{2}+{C}^{2}}\sqrt{{a}^{2}+{b}^{2}+{c}^{2}}}.$$ |

In fact, the right hand side (http://planetmath.org/Equation) is the cosine of the angle $\alpha $ between $l$ and the surface normal of $\tau $ (see angle between two lines), and $\omega $ is the complementary angle^{} of $\alpha $.

Example. Consider the $xy$-plane and the line $l$ through the origin and the point $(1,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}1})$. We can use the components $1,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}1}$ for the direction vector of $l$ and the components $0,\mathrm{\hspace{0.17em}0},\mathrm{\hspace{0.17em}1}$ for the normal vector of the plane. We have

$$\omega =\mathrm{arcsin}\frac{1\cdot 0+1\cdot 0+1\cdot 1}{\sqrt{{1}^{2}+{1}^{2}+{1}^{2}}\sqrt{{0}^{2}+{0}^{2}+{1}^{2}}}=\mathrm{arcsin}\frac{1}{\sqrt{3}}\approx {35.26}^{\circ}.$$ |

Title | angle between line and plane |

Canonical name | AngleBetweenLineAndPlane |

Date of creation | 2013-03-22 17:30:14 |

Last modified on | 2013-03-22 17:30:14 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 13 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 51N20 |

Synonym | slant |

Synonym | inclination |

Related topic | AngleBetweenTwoLines |

Related topic | DotProduct |

Related topic | EquationOfPlane |

Related topic | AngleBetweenTwoPlanes |

Related topic | NormalOfPlane |

Related topic | ProjectionOfRightAngle |

Defines | angle between plane and line |