Ex 11.3, 1 - Chapter 11 Class 12 Three Dimensional Geometry

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 & ﷮ 𝑎﷮2﷯+ 𝑏﷮2﷯+ 𝑐﷮2﷯﷯ = ﷮ 0﷮2﷯+ 0﷮2﷯+ 1﷮2﷯﷯ = 1 l = 𝑎﷮ ﷮ 𝑎﷮2﷯+ 𝑏﷮2﷯+ 𝑐﷮2﷯﷯﷯ , m = 𝑏﷮ ﷮ 𝑎﷮2﷯+ 𝑏﷮2﷯+ 𝑐﷮2﷯﷯﷯ , n = 𝑐﷮ ﷮ 𝑎﷮2﷯+ 𝑏﷮2﷯+ 𝑐﷮2﷯﷯﷯ Direction cosines of the normal to the plane are l = 𝑎﷮ ﷮ 𝑎﷮2﷯+ 𝑏﷮2﷯+ 𝑐﷮2﷯﷯﷯ , m = 𝑏﷮ ﷮ 𝑎﷮2﷯+ 𝑏﷮2﷯+ 𝑐﷮2﷯﷯﷯ , n = 𝑐﷮ ﷮ 𝑎﷮2﷯+ 𝑏﷮2﷯+ 𝑐﷮2﷯﷯﷯ l = 0﷮1﷯ , m = 0﷮1﷯ , n = 1﷮1﷯ l = 0, m = 0, n = 1 ∴ Direction cosines of the normal to the plane are = (0, 0, 1) And, Distance form the origin = 𝑑﷮ ﷮ 𝑎﷮2﷯+ 𝑏﷮2﷯+ 𝑐﷮2﷯﷯﷯ = 2﷮1﷯ = 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 & ﷮ 𝑎﷮2﷯+ 𝑏﷮2﷯+ 𝑐﷮2﷯﷯ = ﷮ 1﷮2﷯+ 1﷮2﷯+ 1﷮2﷯﷯ = ﷮3﷯ Direction cosines of the normal to the plane are l = 𝑎﷮ ﷮ 𝑎﷮2﷯+ 𝑏﷮2﷯+ 𝑐﷮2﷯﷯﷯ , m = 𝑏﷮ ﷮ 𝑎﷮2﷯+ 𝑏﷮2﷯+ 𝑐﷮2﷯﷯﷯ , n = 𝑐﷮ ﷮ 𝑎﷮2﷯+ 𝑏﷮2﷯+ 𝑐﷮2﷯﷯﷯ 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﷮ ﷮ 𝑎﷮2﷯+ 𝑏﷮2﷯+ 𝑐﷮2﷯﷯﷯ = 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 & ﷮ 𝑎﷮2﷯+ 𝑏﷮2﷯+ 𝑐﷮2﷯﷯ = ﷮ 2﷮2﷯ + 3﷮2﷯ + (−1)﷮2﷯﷯ = ﷮4+9+1﷯ = ﷮14﷯ Direction cosines of the normal to the plane are l = 𝑎﷮ ﷮ 𝑎﷮2﷯+ 𝑏﷮2﷯+ 𝑐﷮2﷯﷯﷯ , m = 𝑏﷮ ﷮ 𝑎﷮2﷯+ 𝑏﷮2﷯+ 𝑐﷮2﷯﷯﷯ , n = 𝑐﷮ ﷮ 𝑎﷮2﷯+ 𝑏﷮2﷯+ 𝑐﷮2﷯﷯﷯ 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﷮ ﷮ 𝑎﷮2﷯+ 𝑏﷮2﷯+ 𝑐﷮2﷯﷯﷯ = 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 & ﷮ 𝑎﷮2﷯+ 𝑏﷮2﷯+ 𝑐﷮2﷯﷯ = ﷮ 0﷮2﷯ + (−5)﷮2﷯ + 0﷮2﷯﷯ = ﷮25﷯ = 5 Direction cosines of the normal to the plane are l = 𝑎﷮ ﷮ 𝑎﷮2﷯+ 𝑏﷮2﷯+ 𝑐﷮2﷯﷯﷯ , m = 𝑏﷮ ﷮ 𝑎﷮2﷯+ 𝑏﷮2﷯+ 𝑐﷮2﷯﷯﷯ , n = 𝑐﷮ ﷮ 𝑎﷮2﷯+ 𝑏﷮2﷯+ 𝑐﷮2﷯﷯﷯ l = 0﷮5﷯, m = −5﷮5﷯, n = 0﷮5﷯ ∴ Direction cosines of the normal to the plane are = (0, –1, 0) And, Distance form the origin = 𝑑﷮ ﷮ 𝑎﷮2﷯+ 𝑏﷮2﷯+ 𝑐﷮2﷯﷯﷯ = 𝟖﷮𝟓﷯