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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. (b) 2x + y + 3z โ€“ 2 = 0 and x โ€“ 2y + 5 = 0 Check parallel Two lines with direction ratios ๐ด_1, ๐ต_1, ๐ถ_1 and ๐ด_2, ๐ต_2, ๐ถ_2 are parallel if ๐‘จ_๐Ÿ/๐‘จ_๐Ÿ = ๐‘ฉ_๐Ÿ/๐‘ฉ_๐Ÿ = ๐‘ช_๐Ÿ/๐‘ช_๐Ÿ So, ๐ด_1/๐ด_2 = 2/(โˆ’1) = โˆ’2, ๐ต_1/๐ต_2 = 1/2 , ๐ถ_1/๐ถ_2 = 3/0 Since, direction ratios are not proportional, the two normal are not parallel. โˆด Given two planes are not parallel. Check perpendicular Two lines with direction ratios ๐ด_1, ๐ต_1, ๐ถ_1 and ๐ด_2, ๐ต_2, ๐ถ_2 are perpendicular if ๐‘จ_๐Ÿ ๐‘จ_๐Ÿ + ๐‘ฉ_๐Ÿ ๐‘ฉ_๐Ÿ + ๐‘ช_๐Ÿ ๐‘ช_๐Ÿ = 0 Now, ๐ด_1 ๐ด_2 + ๐ต_1 ๐ต_2 + ๐ถ_1 ๐ถ_2 = (2 ร— โˆ’1) + (1 ร— 2) + (3 ร— 0) = โˆ’2 + 2 + 0 = 0 Since ๐ด_1 ๐ด_2 + ๐ต_1 ๐ต_2 + ๐ถ_1 ๐ถ_2 = 0 The two normal are perpendicular. Since normal are perpendicular, planes are perpendicular.

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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.