If ( l ₁, ~m ₁, n ₁ ), ( l ₂, ~m ₂, n ₂ ) are the direction cosines of two lines, then (1₁ m₂-1₂ m₁ )^2+ (m₁ n₂-m₂ n₁ )^2+ ( n ₁ l ₂- n ₂ l ₁ )^2+ ( l ₁ l ₂+ m ₁ ~m ₂+ n ₁ n ₂ )^2=

Step-by-step solution

Given (l₁, m₁, n ₁ ), (l₂, m₂, n₂ ) are d.cs of two lines. aligned & So l₁^2+m₁^2+n₁^2=1, l₂^2+m₂^2+n₂^2=1 & Now, (l₁ m₂-l₂ m₁ )^2+ (m₁ n₂-m₂ n₁ )^2+ (n₁ l₂-n₂ l₁ )^2+ (l₁ l₂ . & .+m₁ m₂+n₁ n₂ )^2 & =l₁^2 m₂^2+l₂^2 m₁^2-2 l₁ l₂ m₁ m₂+m₁^2 n₂^2+m₂^2 n₁^2 & -2 m₁ m₂ n₁ n₂+n₁^2 l₂^2+n₂^2 l₁^2-2 l₁ l₂ n₁ n₂+ & l₁^2 l₂^2+m₁^2 m₂^2+n₁^2 n₂^2 & +2 l₁ l₂ m₁ m₂+2 m₁ m₂ n₁ n₂+2 l₁ l₂ n₁ n₂ & =l₂^2 (l₁^2+m₁^2+n₁^2 )+m₂^2 (m₁^2+n₁^2+l₁^2 ) & +n₂^2 (l₁^2+m₁^2+n₁^2 )=l₂^2+m₂^2+n₂^2=1 aligned