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

Transcript

Misc 2 If 𝑙_1 , π‘š_1, 𝑛_1 and 𝑙_2 , π‘š_2, 𝑛_2 are the direction cosines of two mutually perpendicular lines, show that direction cosines of line perpendicular to both of these are π‘š_1 𝑛_2 – π‘š_2 𝑛_1 , 𝑛_1 𝑙_2 – 𝑛_2 𝑙_1 , 𝑙_1 π‘š_2 – 𝑙_2 π‘š_1.We know that π‘Ž βƒ— Γ— 𝑏 βƒ— is perpendicular to both π‘Ž βƒ— & 𝑏 βƒ— So, required line is cross product of lines having direction cosines 𝑙_1 , π‘š_1, 𝑛_1 and 𝑙_2 , π‘š_2, 𝑛_2 Required line = |β– 8(𝑖 Μ‚&𝑗 Μ‚&π‘˜ Μ‚@𝑙_1&π‘š_1&𝑛_1@𝑙_2&π‘š_2&𝑛_2 )| = 𝑖 Μ‚ (π‘š_1 𝑛_2 – π‘š_2 𝑛_1) – 𝑗 Μ‚ (𝑙_1 𝑛_2 – 𝑙_2 𝑛_1) + π‘˜ Μ‚(𝑙_1 π‘š_2 – 𝑙_2 π‘š_1) = (π‘š_1 𝑛_2 – π‘š_2 𝑛_1) 𝑖 Μ‚ + (𝑙_2 𝑛_1βˆ’π‘™_1 𝑛_2) 𝑗 Μ‚ + (𝑙_1 π‘š_2 – 𝑙_2 π‘š_1) π‘˜ Μ‚ Hence, direction cosines = π‘š_1 𝑛_2 – π‘š_2 𝑛_1 , 𝑛_1 𝑙_2 – 𝑛_2 𝑙_1 , 𝑙_1 π‘š_2 – 𝑙_2 π‘š_1 ∴ Direction cosines of the line perpendicular to both of these are π‘š_1 𝑛_2 – π‘š_2 𝑛_1 , 𝑛_1 𝑙_2 – 𝑛_2 𝑙_1 , 𝑙_1 π‘š_2 – 𝑙_2 π‘š_1. Hence proved

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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.