The acute angle between the lines whose direction cosines are given by the equations l+m+n=0 and 2 l m+2 l n-m n=0 is

Step-by-step solution

Given equations are, On squaring both side of the Eq. (i). array cc & l^2+m^2+n^2+2 l m+2 m n+2 n l=0 & l^2+m^2+n^2=1, array From Eqs. (ii) and (iv), 3 m n=-1 m n=- 1 3 From Eq. (iii) 6 l^2-1=0 , let root of this equation is l₁ and l₂ , so l₁ l₂=- 1 6 . Now, from Eqs. (i) and (ii), 2 l m+(2 l-m)(-l-m)=0 array rlrl & 2 l^2-l m-m^2 =0 & 2 ( l m )^2- ( l m )-1 =0 & l₁ l₂ m₁ m₂ = -1 2 & l₁ l₂ -1 / 6 = m₁ m₂ 2 / 6 & Similarly, l₁ l₂ -1 / 6 = n₁ n₂ 2 / 6 array So, = |l₁ l₂+m₁ m₂+n₁ n₂ |= 3 6 = 1 2 = 3