Angle between two planes
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 l1 and l2, which lie on planes and which is perpendicular to lines of planes crossing.
Angle between two planes formulas
If A1x + B1y + C1z + D1 = 0 and A2x + B2y + C2z + D2 = 0 are a plane equations, then angle between planes can be found using the following formula
|cos α =|||A1·A2 + B1·B2 + C1·C2||
|√A12 + B12 + C12√A22 + B22 + C22|
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|
|√22 + 42 + (-4)2√42 + 32 + 02||√36√25||30||3|
|Answer: the cosine of the angle between the planes is
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