The angle between lines whose direction cosines satisfy the equation +m+n=0 and ^2-m^2-n^2=0 , is

Step-by-step solution

The direction cosines ( , m, n) satisfy + m + n = 0 and ^2 - m^2 - n^2 = 0 . From the first equation, n = -( + m) . Substituting into the second equation: ^2 - m^2 - ( +m)^2 = 0 -2m^2 - 2 m = 0 -2m(m+ ) = 0 Two cases emerge from this equation. Case 1: m = 0 From + m + n = 0 and m = 0 , we find = -n . Using the identity ^2 + m^2 + n^2 = 1 yields 2n^2 = 1 , so n = 1 2 . One solution gives direction cosines (- 1 2 , 0, 1 2 ) . Case 2: = -m From + m + n = 0 and = -m , we find n = 0 . The identity becomes 2m^2 = 1 , so