The acute angle between two lines such that the direction cosines l, m, n , of each of them satisfy the equations l+m+n=0 and l^2+m^2-n^2=0 is :

Step-by-step solution

Let l₁, m₁, n₁ and l₂, m₂, n₂ be the d.c of line 1 and 2 respectively, then as given l₁+m₁+n₁=0 and l₂+m₂+n₂=0 and l₁ ^2+m₁ ^2-n₁^2=0 and l₂^2+m₂^2-n₂^2=0 ( l+m+n=0 . and .l^2+m^2-n^2=0 ) Angle between lines, is =l₁ l₂+m₁ m₂+n₁ n₂ As given l^2+m^2=n^2 and l+m=-n (-n)^2-2 l m=n^2 2 l m=0 or l m=0 So l₁ m₁=0, l₂ m₂=0 If l₁=0, m₁ 0 then l₁ m₂=0 If m₁=0, l₁ 0 then l₂ m₁=0 If l₂=0, m₂ 0 then l₂ m₁=0 If m₂=0, l₂ 0 then l₁ m₂=0 Also l₁ l₂=0 and m₁ m₂=0 aligned & l^2+m^2-n^2=l^2+m^2+n^2-2 n^2=0 & 1-2 n^2=0 n= 1 2 & n₁= 1 2