Angle between two planes

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  â‹…  1  = p₁ and  â‹…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 a2x + b2y + c2z + d2 = 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 1  = 2ˆi + 2ˆ j + 2ˆk and  2  = 4ˆi - 2ˆ j + 2ˆk  respectively.

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