Maharashtra State BoardHSC Arts 12th Board Exam
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Angle Between Two Planes

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The angle between two planes is defined as the angle between their normals Fig,


Observe that if θ is an angle between the two planes, then so is 180 – θ Fig.

If `vec n _1` and `vec n_2` are normals to the planes and θ be the angle between the planes  
`vec r . vec n _1 = d_1` and `vec r . vec n _2 = d_2` . 
Then θ is the angle between the normals to the planes drawn from some common point We have  
cos θ = `|(vec n_1 . vec n_2)/ (|vec n _1| |vec n_2|)|`

Cartesian form 
Let θ be the angle between the planes, 
`A_1x + B_1y +C_1z + D_1 = 0` and `A_2x +B_2y + C_2 z + D_2 = 0`
The direction ratios of the normal to the planes are `A_1, B_1, C_1` and `A_2, B_2, C_2` respectively.
Therefore , cos θ = `|(A_1 A_2 + B_1 B_2 + C_1 C_2)/ (sqrt(A_1^2 + B_1^2 + C_1^2 ) sqrt (A_2^2 + B_2^2 +C_2^2))|`

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Angle Between Two Planes [00:02:45]
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