Question
The angle between the lines whose direction cosines satisfy the equations $$l + m + n = 0$$ and $${l^2} = {m^2} + {n^2}$$ is :
A.
$$\frac{\pi }{6}$$
B.
$$\frac{\pi }{2}$$
C.
$$\frac{\pi }{3}$$
D.
$$\frac{\pi }{4}$$
Answer :
$$\frac{\pi }{3}$$
Solution :
$$\eqalign{
& {\left\{ { - \left( {m + n} \right)} \right\}^2} = {m^2} + {n^2}\,\,\,\,\, \Rightarrow mn = 0 \cr
& {\text{When }}m = 0,\,l = - n{\text{ direction ratios are }} - 1,\,0,\,1 \cr
& {\text{When }}n = 0,\,l = - m{\text{ direction ratios are }} - 1,\,1,\,0 \cr
& \therefore \,\cos \,\theta = \frac{{ - 1}}{{\sqrt 2 }}.\frac{{ - 1}}{{\sqrt 2 }} + \frac{0}{{\sqrt 2 }}.\frac{1}{{\sqrt 2 }} + \frac{1}{{\sqrt 2 }}.\frac{0}{{\sqrt 2 }} = \frac{1}{2} \cr} $$