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Definition, Theorem, Proof, Solved Example Problems, Solution - Angle between two planes | 12th Mathematics : UNIT 6 : Applications of Vector Algebra

Chapter: 12th Mathematics : UNIT 6 : Applications of Vector Algebra

Angle between two planes

The angle between two given planes is same as the angle between their normals.

Angle between two planes

The angle between two given planes is same as the angle between their normals.

 


Proof

If θ is the acute angle between two planes    1  = p1 and  2  = p2  , then θ is the acute angle between their normal vectors 1 and 2 

Therefore,


Remark


 

Theorem 6.19

The acute angle θ between the planes a1x + b1y + c1z + d1 = 0 and a2b2c2d2 = 0 is given by 


Proof

If 1 and 2 are the vectors normal to the two given planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 respectively. Then, 

Therefore, using equation (1) in theorem 6.18 the acute angle θ between the planes is given by


Remark

(i) The planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 are perpendicular if a1a2  + b1b2   + c1c2  = 0

(ii) The planes a1x + b1 y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 are parallel if 

(iii) Equation of a plane parallel to the plane ax + by + cz = p is ax + by + cz = k , k R.

 

Example 6.47

Find the acute angle between the planes .(2 ˆi + 2ˆ j + 2ˆk ) = 11 and 4x - 2 y + 2z = 15

Solution

The normal vectors of the two given planes  = (2 ˆi + 2ˆ j + 2ˆk ) = 11 and 4x - 2 y + 2z = 15 are  = 2ˆi + 2ˆ j + 2ˆk and   = 4ˆi - 2ˆ j + 2ˆk  respectively.

If θ is the acute angle between the planes, then we have


 

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12th Mathematics : UNIT 6 : Applications of Vector Algebra : Angle between two planes | Definition, Theorem, Proof, Solved Example Problems, Solution

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12th Mathematics : UNIT 6 : Applications of Vector Algebra


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