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Last updated at Feb. 1, 2020 by Teachoo
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Example 21 Show that the lines (๐ฅ + 3)/( โ3) = (๐ฆ โ 1)/1 = (๐ง โ 5)/5 and (๐ฅ + 1)/( โ1) = (๐ฆ โ 2)/2 = (๐ง โ 5)/5 are coplanar. Two lines (๐ฅ โ ๐ฅ_1)/๐_1 = (๐ฆ โ ๐ฆ_1)/๐_1 = (๐ง โ ๐ง_1)/๐_1 and (๐ฅ โ ๐ฅ_2)/๐_2 = (๐ฆ โ ๐ฆ_2)/๐_2 = (๐ง โ ๐ง_2)/๐_2 are coplanar if |โ 8(๐_๐โ๐_๐&๐_๐โ๐_๐&๐_๐โ๐_๐@๐_๐&๐_๐&๐_๐@๐_๐&๐_๐&๐_๐ )| = 0 Given, the two lines are Given, (๐ + ๐)/( โ ๐) = (๐ โ ๐)/๐ = (๐ โ ๐)/๐ (๐ฅ โ(โ3))/( โ 3) = (๐ฆ โ1)/1 = (๐งโ 5)/5 Comparing (๐ฅ โ ๐ฅ_1)/๐_1 = (๐ฆ โ ๐ฆ_1)/๐_1 = (๐ง โ ๐ง_1)/๐_1 ๐ฅ_1 = โ3, ๐ฆ_1 = 1, ๐ง_1= 5 & ๐_1 = โ3, ๐_1 = 1, ๐_1= 5 Given, (๐ + ๐)/( โ ๐) = (๐ โ ๐)/๐ = (๐ โ ๐)/๐ (๐ฅ โ (โ1))/( โ 1) = (๐ฆ โ 2)/2 = (๐ง โ 5)/5 Comparing (๐ฅ โ ๐ฅ_2)/๐_2 = (๐ฆ โ ๐ฆ_2)/๐_2 = (๐ง โ ๐ง_2)/๐_2 ๐ฅ_2 = โ1, ๐ฆ_2 = 2, ๐ง_2= 5 & ๐_2 = โ1, ๐_2 = 2, ๐_2= 5 Now, |โ 8(๐ฅ_2โ๐ฅ_1&๐ฆ_2โ๐ฆ_1&๐ง_2โ๐ง_1@๐_1&๐_1&๐_1@๐_2&๐_2&๐_2 )| `= |โ 8( โ1โ(โ3)&2โ1&5โ5@ โ3&1&5@ โ1&2&5)| = |โ 8(2&1&0@โ3&1&5@โ1&2&5)| = 2[(1ร5)โ(2ร5)] โ 1[(โ3ร5)โ(โ 1ร5)] + 0 [(โ3ร2)โ(โ1ร1)] = 2[5โ10]โ 1[โ 15โ(โ 5)] + 0 = 2(โ5) โ1(โ10) = โ10 + 10 = 0 Therefore, the given two lines are coplanar.
Examples
Example, 2 Important
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Example, 4 Important
Example, 5 Important
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Example 8
Example, 9 Deleted for CBSE Board 2021 Exams only
Example 10 Deleted for CBSE Board 2021 Exams only
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Example 12 Important
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Example 20 Important
Example 21 Important You are here
Example 22 Deleted for CBSE Board 2021 Exams only
Example 23 Important Deleted for CBSE Board 2021 Exams only
Example 24
Example, 25 Important
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Example 30 Important
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