Example 21 - Show that lines x+3/3 = y-1/1 = z-5/5 are coplanar

Example 21 - Chapter 11 Class 12 Three Dimensional Geometry - Part 2
Example 21 - Chapter 11 Class 12 Three Dimensional Geometry - Part 3

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Question 11 Show that the lines (๐‘ฅ + 3)/( โˆ’3) = (๐‘ฆ โˆ’ 1)/1 = (๐‘ง โˆ’ 5)/5 and (๐‘ฅ + 1)/( โˆ’1) = (๐‘ฆ โˆ’ 2)/2 = (๐‘ง โˆ’ 5)/5 are coplanar. Two lines (๐‘ฅ โˆ’ ๐‘ฅ_1)/๐‘Ž_1 = (๐‘ฆ โˆ’ ๐‘ฆ_1)/๐‘_1 = (๐‘ง โˆ’ ๐‘ง_1)/๐‘_1 and (๐‘ฅ โˆ’ ๐‘ฅ_2)/๐‘Ž_2 = (๐‘ฆ โˆ’ ๐‘ฆ_2)/๐‘_2 = (๐‘ง โˆ’ ๐‘ง_2)/๐‘_2 are coplanar if |โ– 8(๐’™_๐Ÿโˆ’๐’™_๐Ÿ&๐’š_๐Ÿโˆ’๐’š_๐Ÿ&๐’›_๐Ÿโˆ’๐’›_๐Ÿ@๐’‚_๐Ÿ&๐’ƒ_๐Ÿ&๐’„_๐Ÿ@๐’‚_๐Ÿ&๐’ƒ_๐Ÿ&๐’„_๐Ÿ )| = 0 Given, the two lines are Given, (๐’™ + ๐Ÿ‘)/( โˆ’ ๐Ÿ‘) = (๐’š โˆ’ ๐Ÿ)/๐Ÿ = (๐’› โˆ’ ๐Ÿ“)/๐Ÿ“ (๐‘ฅ โˆ’(โˆ’3))/( โˆ’ 3) = (๐‘ฆ โˆ’1)/1 = (๐‘งโˆ’ 5)/5 Comparing (๐‘ฅ โˆ’ ๐‘ฅ_1)/๐‘Ž_1 = (๐‘ฆ โˆ’ ๐‘ฆ_1)/๐‘_1 = (๐‘ง โˆ’ ๐‘ง_1)/๐‘_1 ๐‘ฅ_1 = โˆ’3, ๐‘ฆ_1 = 1, ๐‘ง_1= 5 & ๐‘Ž_1 = โˆ’3, ๐‘_1 = 1, ๐‘_1= 5 Given, (๐’™ + ๐Ÿ)/( โˆ’ ๐Ÿ) = (๐’š โˆ’ ๐Ÿ)/๐Ÿ = (๐’› โˆ’ ๐Ÿ“)/๐Ÿ“ (๐‘ฅ โˆ’ (โˆ’1))/( โˆ’ 1) = (๐‘ฆ โˆ’ 2)/2 = (๐‘ง โˆ’ 5)/5 Comparing (๐‘ฅ โˆ’ ๐‘ฅ_2)/๐‘Ž_2 = (๐‘ฆ โˆ’ ๐‘ฆ_2)/๐‘_2 = (๐‘ง โˆ’ ๐‘ง_2)/๐‘_2 ๐‘ฅ_2 = โˆ’1, ๐‘ฆ_2 = 2, ๐‘ง_2= 5 & ๐‘Ž_2 = โˆ’1, ๐‘_2 = 2, ๐‘_2= 5 Now, |โ– 8(๐‘ฅ_2โˆ’๐‘ฅ_1&๐‘ฆ_2โˆ’๐‘ฆ_1&๐‘ง_2โˆ’๐‘ง_1@๐‘Ž_1&๐‘_1&๐‘_1@๐‘Ž_2&๐‘_2&๐‘_2 )| `= |โ– 8( โˆ’1โˆ’(โˆ’3)&2โˆ’1&5โˆ’5@ โˆ’3&1&5@ โˆ’1&2&5)| = |โ– 8(2&1&0@โˆ’3&1&5@โˆ’1&2&5)| = 2[(1ร—5)โˆ’(2ร—5)] โˆ’ 1[(โˆ’3ร—5)โˆ’(โˆ’ 1ร—5)] + 0 [(โˆ’3ร—2)โˆ’(โˆ’1ร—1)] = 2[5โˆ’10]โˆ’ 1[โˆ’ 15โˆ’(โˆ’ 5)] + 0 = 2(โˆ’5) โˆ’1(โˆ’10) = โˆ’10 + 10 = 0 Therefore, the given two lines are coplanar.

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.