. . Angle between two planes The angle between two given planes is same as the angle between their normals. Theorem .
The acute angle θ between the two planes r n and r n is given by n n n n − Proof If θ is the acute angle between two planes r n and r n , then θ is the acute angle between their normal vectors n and n . Therefore, n n n n n n n n ⇒ ... ( ) Remark (i) If two planes r n r n are perpendicular, then n n (ii) If the planes r n r n are parallel, then , where λ is a scalar (iii) Equation of a plane parallel to the plane r n is r n k k ∈ Theorem . The acute angle θ between the planes a x b y c z a x b y c z is given by a a b b c c − Fig.
. n ° − θ r n n r n Vector - - Applications of Vector Algebra Proof If n n are the vectors normal to the two given planes a x b y c z a x b y c z respectively. Then, ˆ a i b j c k ˆ a i b j c k Therefore, using equation ( ) in theorem . the acute angle θ between the planes is given by a a b b c c − Remark (i) The planes a x b y c z a x b y c z are perpendicular if a a b b c c (ii) The planes a x b y c z a x b y c z are parallel if (iii) Equation of a plane parallel to the plane ax by cz is ax by cz , k ∈ Example .
Find the acute angle between the planes