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  1. Chapter 11 Class 12 Three Dimensional Geometry
  2. Serial order wise

Transcript

Ex 11.3, 13 In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them. (a) 7x + 5y + 6z + 30 = 0 and 3x – y – 10z + 4 = 0 7x + 5y + 6z + 30 = 0 7x + 5y + 6z = −30 −(7x + 5y + 6z) = 30 −7x – 5y – 6z = 30 Comparing with A1x + B1y + C1z = d1 Direction ratios of normal = –7, –5, –6 A1 = –7 , B1 = –5 , C1 = –6 3x − y − 10z + 4 = 0 3x − y − 10z = −4 −(3x − y − 10z) = 4 −3x + y + 10z = 4 Comparing with A2x + B2y + C2z = d2 Direction ratios of normal = –3, 1, 10 A2 = –3 , B2 = 1 , C2 = 10 Check parallel Two lines with direction ratios 𝐴_1, 𝐵_1, 𝐶_1 and 𝐴_2, 𝐵_2, 𝐶_2 are parallel if 𝑨_𝟏/𝑨_𝟐 = 𝑩_𝟏/𝑩_𝟐 = 𝑪_𝟏/𝑪_𝟐 So, 𝐴_1/𝐴_2 = (−7)/(−3) = 7/3, 𝐵_1/𝐵_2 = (−5)/1 = −5 , 𝐶_1/𝐶_2 = (−6)/10 = (−3)/5 Since 𝐴_1/𝐴_2 ≠ 𝐵_1/𝐵_2 ≠ 𝐶_1/𝐶_2 So, the two normal are not parallel. ∴ Given two planes are not parallel. Check perpendicular Two lines with direction ratios 𝐴_1, 𝐵_1, 𝐶_1 and 𝐴_2, 𝐵_2, 𝐶_2 are perpendicular if 𝑨_𝟏 𝑨_𝟐 + 𝑩_𝟏 𝑩_𝟐 + 𝑪_𝟏 𝑪_𝟐 = 0 𝐴_1 𝐴_2 + 𝐵_1 𝐵_2 + 𝐶_1 𝐶_2 = (−7 × −3) + (−5 × 1) + (−6 × 10) = 21 + (−5) + (−60) = −44 Since, 𝐴_1 𝐴_2 + 𝐵_1 𝐵_2 + 𝐶_1 𝐶_2 ≠ 0 Therefore, the two normal are not perpendicular. Hence, the given two planes are not perpendicular. Finding angle Now, the angle between two planes 𝐴_1x + 𝐵_1 𝑦 + 𝐶_1 𝑧 = d1 and 𝐴_2x + 𝐵_2 𝑦 + 𝐶_2 𝑧 = d2 is given by cos θ = |(𝑨_𝟏 𝑨_𝟐 + 𝑩_𝟏 𝑩_𝟐 + 𝑪_𝟏 𝑪_𝟐)/(√(〖𝑨_𝟏〗^𝟐 + 〖𝑩_𝟏〗^𝟐 + 〖𝑪_𝟏〗^𝟐 ) √(〖𝑨_𝟐〗^𝟐 + 〖𝑩_𝟐〗^𝟐 + 〖𝑪_𝟐〗^𝟐 ))| = |((−7 × −3) + (−5 × 1) + (−6 × 10) )/(√((−7)^2+(−5)^2+(−6)^2 ) √((−3)^2 + 1^2 + 〖10〗^2 ))| = |(21 − 5 − 60)/(√(49 + 25 + 36) √(9 + 1 + 100))| = |(−44)/(√110 √110)| = |(−44)/110| = |(−2)/5| = 2/5 Hence, cos θ = 2/5 ∴ θ = 〖𝐜𝐨𝐬〗^(−𝟏)⁡(𝟐/𝟓) Hence, angle between two planes is cos^(−1)⁡(2/5) Ex 11.3, 13 In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them. (b) 2x + y + 3z – 2 = 0 and x – 2y + 5 = 0 Check parallel Two lines with direction ratios 𝐴_1, 𝐵_1, 𝐶_1 and 𝐴_2, 𝐵_2, 𝐶_2 are parallel if 𝑨_𝟏/𝑨_𝟐 = 𝑩_𝟏/𝑩_𝟐 = 𝑪_𝟏/𝑪_𝟐 So, 𝐴_1/𝐴_2 = 2/(−1) = −2, 𝐵_1/𝐵_2 = 1/2 , 𝐶_1/𝐶_2 = 3/0 Since, direction ratios are not proportional, the two normal are not parallel. ∴ Given two planes are not parallel. Check perpendicular Two lines with direction ratios 𝐴_1, 𝐵_1, 𝐶_1 and 𝐴_2, 𝐵_2, 𝐶_2 are perpendicular if 𝑨_𝟏 𝑨_𝟐 + 𝑩_𝟏 𝑩_𝟐 + 𝑪_𝟏 𝑪_𝟐 = 0 Now, 𝐴_1 𝐴_2 + 𝐵_1 𝐵_2 + 𝐶_1 𝐶_2 = (2 × −1) + (1 × 2) + (3 × 0) = −2 + 2 + 0 = 0 Since 𝐴_1 𝐴_2 + 𝐵_1 𝐵_2 + 𝐶_1 𝐶_2 = 0 The two normal are perpendicular. Since normal are perpendicular, planes are perpendicular. Ex 11.3, 13 In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them. (c) 2x – 2y + 4z + 5 = 0 and 3x – 3y + 6z – 1 = 0 2x − 2y + 4z + 5 = 0 2x − 2y + 4z = −5 −2x + 2y − 4z = 5 Comparing with A1x + B1y + C1z = d1 Direction ratios of normal = –2, 2, –4 A1 = –2 , B1 = 2 , C1 = –4 3x − 3y + 6z − 1 = 0 3x − 3y + 6z = 1 Comparing with A2x + B2y + C2z = d2 Direction ratios of normal = 3, –3, 6 A2 = 3 , B2 = –3 , C2 = 6 Check parallel Two lines with direction ratios 𝐴_1, 𝐵_1, 𝐶_1 and 𝐴_2, 𝐵_2, 𝐶_2 are parallel if 𝑨_𝟏/𝑨_𝟐 = 𝑩_𝟏/𝑩_𝟐 = 𝑪_𝟏/𝑪_𝟐 Here, 𝐴_1/𝐴_2 = (−2)/3, 𝐵_1/𝐵_2 = 2/(−3) = (−2)/3 , 𝐶_1/𝐶_2 = (−4)/6 = (−2)/3 Since, 𝐴_1/𝐴_2 = 𝐵_1/𝐵_2 = 𝐶_1/𝐶_2 = (−2)/3 Therefore, the two normal are parallel Since normal are parallel, the two planes are parallel. Ex 11.3, 13 In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them. (d) 2x – y + 3z – 1 = 0 and 2x – y + 3z + 3 = 0Given, the two planes are 2x – y + 3z – 1 = 0 2x – y + 3z = 1 Comparing with A1x + B1y + C1z = d1 Direction ratios of normal = 2, –1, 3 A1 = 2 , B1 = –1 , C1 = 3 2x – y + 3z + 3 = 0 2x – y + 3z = –3 –2x + y – 3z = 3 Comparing with A2x + B2y + C2z = d2 Direction ratios of normal = –2, 1, –3 A2 = –2 , B2 = 1 , C2 = –3 Check parallel Two lines with direction ratios 𝐴_1, 𝐵_1, 𝐶_1 and 𝐴_2, 𝐵_2, 𝐶_2 are parallel if 𝑨_𝟏/𝑨_𝟐 = 𝑩_𝟏/𝑩_𝟐 = 𝑪_𝟏/𝑪_𝟐 Here, 𝐴_1/𝐴_2 = 2/(−2) = −1, 𝐵_1/𝐵_2 = (−1)/1 = −1, 𝐶_1/𝐶_2 = 3/(−3) = −1 Since 𝐴_1/𝐴_2 = 𝐵_1/𝐵_2 = 𝐶_1/𝐶_2 = −1, Therefore, the two normal are parallel Since normal are parallel, the two planes are parallel. Ex 11.3, 13 In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them. (e) 4x + 8y + z – 8 = 0 and y + z − 4 = 0 4x + 8y + z − 8 = 0 4x + 8y + z = 8 4x + 8y + 1z = 8 Comparing with A1x + B1y + C1z = d1 Direction ratios of normal = 4, 8, 1 A1 = 4, B1 = 8, C1 = 1 y + z − 4 = 0 y + z = 4 0x + 1y + 1z = 4 Comparing with A2x + B2y + C2z = d2 Direction ratios of normal = 0, 1, 1 A2 = 0, B2 = 1, C2 = 1 Check parallel Two lines with direction ratios 𝐴_1, 𝐵_1, 𝐶_1 and 𝐴_2, 𝐵_2, 𝐶_2 are parallel if 𝑨_𝟏/𝑨_𝟐 = 𝑩_𝟏/𝑩_𝟐 = 𝑪_𝟏/𝑪_𝟐 So, 𝐴_1/𝐴_2 = 4/0 = −1, 𝐵_1/𝐵_2 = 8/1 = 8 , 𝐶_1/𝐶_2 = 1/1 = 1 Since 𝐴_1/𝐴_2 ≠ 𝐵_1/𝐵_2 ≠ 𝐶_1/𝐶_2 So, the two normal are not parallel. ∴ Given two planes are not parallel. Check perpendicular Two lines with direction ratios 𝐴_1, 𝐵_1, 𝐶_1 & 𝐴_2, 𝐵_2, 𝐶_2 are perpendicular if 𝑨_𝟏 𝑨_𝟐 + 𝑩_𝟏 𝑩_𝟐 + 𝑪_𝟏 𝑪_𝟐 = 0 𝐴_1 𝐴_2 + 𝐵_1 𝐵_2 + 𝐶_1 𝐶_2 = (4 × 0) + (8 × 1) + (1 × 1) = 0 + 8 + 1 = 9 Since, 𝐴_1 𝐴_2 + 𝐵_1 𝐵_2 + 𝐶_1 𝐶_2 ≠ 0 Therefore, the two normal are not perpendicular. Hence, the given two planes are not perpendicular. Finding angle Now, the angle between two planes 𝐴_1x + 𝐵_1 𝑦 + 𝐶_1 𝑧 = d1 and 𝐴_2x + 𝐵_2 𝑦 + 𝐶_2 𝑧 = d2 is given by cos θ = |(𝑨_𝟏 𝑨_𝟐 + 𝑩_𝟏 𝑩_𝟐 + 𝑪_𝟏 𝑪_𝟐)/(√(〖𝑨_𝟏〗^𝟐 + 〖𝑩_𝟏〗^𝟐 + 〖𝑪_𝟏〗^𝟐 ) √(〖𝑨_𝟐〗^𝟐 + 〖𝑩_𝟐〗^𝟐 + 〖𝑪_𝟐〗^𝟐 ))| = |((4 × 0) + (8 × 1) + (1 × 1) )/(√(4^2 + 8^2 + 1^2 ) √(0^2 + 1^2 + 1^2 ))| = |(0 + 8 + 1 )/(√(16 + 64 +1 ) √(0+1+1))| = |9/(√(81 ) √2)| = 9/(9√2) = 1/√2 So, cos θ = 1/√2 ∴ θ = 45° Therefore, the angle between the given two planes is 45°

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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.