Question 2 - Angle between two lines - Direction ratios or cosines - Chapter 11 Class 12 Three Dimensional Geometry

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Question 2 If 𝑙_1 , 𝑚_1, 𝑛_1 and 𝑙_2 , 𝑚_2, 𝑛_2 are the direction cosines of two mutually perpendicular lines, show that direction cosines of line perpendicular to both of these are 𝑚_1 𝑛_2 – 𝑚_2 𝑛_1 , 𝑛_1 𝑙_2 – 𝑛_2 𝑙_1 , 𝑙_1 𝑚_2 – 𝑙_2 𝑚_1.We know that 𝑎 ⃗ × 𝑏 ⃗ is perpendicular to both 𝑎 ⃗ & 𝑏 ⃗ So, required line is cross product of lines having direction cosines 𝑙_1 , 𝑚_1, 𝑛_1 and 𝑙_2 , 𝑚_2, 𝑛_2 Required line = |■8(𝑖 ̂&𝑗 ̂&𝑘 ̂@𝑙_1&𝑚_1&𝑛_1@𝑙_2&𝑚_2&𝑛_2 )| = 𝑖 ̂ (𝑚_1 𝑛_2 – 𝑚_2 𝑛_1) – 𝑗 ̂ (𝑙_1 𝑛_2 – 𝑙_2 𝑛_1) + 𝑘 ̂(𝑙_1 𝑚_2 – 𝑙_2 𝑚_1) = (𝑚_1 𝑛_2 – 𝑚_2 𝑛_1) 𝑖 ̂ + (𝑙_2 𝑛_1−𝑙_1 𝑛_2) 𝑗 ̂ + (𝑙_1 𝑚_2 – 𝑙_2 𝑚_1) 𝑘 ̂ Hence, direction cosines = 𝑚_1 𝑛_2 – 𝑚_2 𝑛_1 , 𝑛_1 𝑙_2 – 𝑛_2 𝑙_1 , 𝑙_1 𝑚_2 – 𝑙_2 𝑚_1 ∴ Direction cosines of the line perpendicular to both of these are 𝑚_1 𝑛_2 – 𝑚_2 𝑛_1 , 𝑛_1 𝑙_2 – 𝑛_2 𝑙_1 , 𝑙_1 𝑚_2 – 𝑙_2 𝑚_1. Hence proved