# Angle between two planes

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Definition.

**The angle between planes**is equal to a angle between their normal vectors.

Definition.

**The angle between planes**is equal to a angle between lines l

_{1}and l

_{2}, which lie on planes and which is perpendicular to lines of planes crossing.

## Angle between two planes formulas

If A_{1}x + B_{1}y + C_{1}z + D_{1} = 0 and A_{2}x + B_{2}y + C_{2}z + D_{2} = 0 are a plane equations, then angle between planes can be found using the following formula

cos α = | |A_{1}·A_{2} + B_{1}·B_{2} + C_{1}·C_{2}| |

√A_{1}^{2} + B_{1}^{2} + C_{1}^{2}√A_{2}^{2} + B_{2}^{2} + C_{2}^{2} |

## Examples of tasks with angle between two planes

Example 1.

To find an Angle between planes 2x + 4y - 4z - 6 = 0 and 4x + 3y + 9 = 0.
**Solution.** Let's use the formula:

cos α = | |2·4 + 4·3 + (-4)·0| | = | |8 + 12| | = | 20 | = | 2 |

√2^{2} + 4^{2} + (-4)^{2}√4^{2} + 3^{2} + 0^{2} |
√36√25 | 30 | 3 |

Answer: the cosine of the angle between the planes is cos α = |
2 | . |

3 |

Analytic geometry: Introduction and contentsDistance between two pointsMidpoint. Coordinates of midpointEquation of a lineEquation of a planeDistance from point to planeDistance between two planesDistance from a point to a line - 2-DimensionalDistance from a point to a line - 3-DimensionalAngle between two planesAngle between line and plane

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