Last updated at Dec. 24, 2019 by Teachoo

Transcript

Ex 11.3, 1 In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin. (a) z = 2 Given equation of plane is z = 2 0x + 0y + 1z = 2 Comparing with ax + by + cz = d a = 0, b = 0, c = 1 & d = 2 & 𝑎2+ 𝑏2+ 𝑐2 = 02+ 02+ 12 = 1 l = 𝑎 𝑎2+ 𝑏2+ 𝑐2 , m = 𝑏 𝑎2+ 𝑏2+ 𝑐2 , n = 𝑐 𝑎2+ 𝑏2+ 𝑐2 Direction cosines of the normal to the plane are l = 𝑎 𝑎2+ 𝑏2+ 𝑐2 , m = 𝑏 𝑎2+ 𝑏2+ 𝑐2 , n = 𝑐 𝑎2+ 𝑏2+ 𝑐2 l = 01 , m = 01 , n = 11 l = 0, m = 0, n = 1 ∴ Direction cosines of the normal to the plane are = (0, 0, 1) And, Distance form the origin = 𝑑 𝑎2+ 𝑏2+ 𝑐2 = 21 = 2. Ex 11.3, 1 In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin. (b) x + y + z = 1 Given equation of plane is x + y + z = 1 1x + 1y + 1z = 1 Comparing with ax + by + cz = d a = 1, b = 1, c = 1 & d = 1 & 𝑎2+ 𝑏2+ 𝑐2 = 12+ 12+ 12 = 3 Direction cosines of the normal to the plane are l = 𝑎 𝑎2+ 𝑏2+ 𝑐2 , m = 𝑏 𝑎2+ 𝑏2+ 𝑐2 , n = 𝑐 𝑎2+ 𝑏2+ 𝑐2 l = 1 3 , m = 1 3, n = 1 3 ∴ Direction cosines of the normal to the plane are = ( 𝟏 𝟑, 𝟏 𝟑, 𝟏 𝟑) And, Distance form the origin = 𝑑 𝑎2+ 𝑏2+ 𝑐2 = 𝟏 𝟑 Ex 11.3, 1 In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin. (c) 2x + 3y – z = 5 Given equation of plane is 2x + 3y − z = 5 2x + 3y − 1z = 5 Comparing with ax + by + cz = d a = 2, b = 3, c = –1 & d = 5 & 𝑎2+ 𝑏2+ 𝑐2 = 22 + 32 + (−1)2 = 4+9+1 = 14 Direction cosines of the normal to the plane are l = 𝑎 𝑎2+ 𝑏2+ 𝑐2 , m = 𝑏 𝑎2+ 𝑏2+ 𝑐2 , n = 𝑐 𝑎2+ 𝑏2+ 𝑐2 l = 2 14, m = 3 14, n = − 1 14 ∴ Direction cosines of the normal to the plane are = ( 𝟐 𝟏𝟒, 𝟑 𝟏𝟒, −𝟏 𝟏𝟒) And, Distance form the origin = 𝑑 𝑎2+ 𝑏2+ 𝑐2 = 𝟓 𝟏𝟒 Ex 11.3, 1 In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin. (d) 5y + 8 = 0 0x − 5y + 0z = 8 Comparing with ax + by + cz = d a = 0, b = –5, c = 0 & d = 8 & 𝑎2+ 𝑏2+ 𝑐2 = 02 + (−5)2 + 02 = 25 = 5 Direction cosines of the normal to the plane are l = 𝑎 𝑎2+ 𝑏2+ 𝑐2 , m = 𝑏 𝑎2+ 𝑏2+ 𝑐2 , n = 𝑐 𝑎2+ 𝑏2+ 𝑐2 l = 05, m = −55, n = 05 ∴ Direction cosines of the normal to the plane are = (0, –1, 0) And, Distance form the origin = 𝑑 𝑎2+ 𝑏2+ 𝑐2 = 𝟖𝟓

Chapter 11 Class 12 Three Dimensional Geometry

Serial order wise

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.