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.

If *Î¸ *is the acute angle between two planes * *â‹… 1 = *p*_{1} and * *â‹…2 = *p*2 , then Î¸ is the acute angle between their
normal vectors _{1} and 2

Therefore,

**Remark**

The acute angle *Î¸ *between the planes *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 is given by

If _{1 }and 2 are the vectors normal to the two
given planes 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 respectively. Then,

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

(i) The planes 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 perpendicular if a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0

(ii) The planes 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 parallel if

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

Find the acute angle between the planes .(2 Ë†*i* + 2Ë† *j* + 2Ë†*k* ) = 11 and 4x - 2 y + 2z = 15

The normal vectors of the two given planes = (2 Ë†*i* + 2Ë† *j* + 2Ë†*k* ) = 11 and 4x - 2
y + 2z = 15 are 1 = 2Ë†*i*
+ 2Ë† *j* + 2Ë†*k* and 2 = 4Ë†*i* - 2Ë† *j* + 2Ë†*k* respectively.

If Î¸ is the acute angle between the planes, then we have

Tags : Definition, Theorem, Proof, Solved Example Problems, Solution , 12th Mathematics : UNIT 6 : Applications of Vector Algebra

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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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