Angle between lines whose direction cosine satisfy $l + m + n = 0, l^{2} + m^{2} - n^{2} = 0$ .

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Solution

$l + m + n = 0 - (i) l^{2} + m^{2} + n^{2} = 1 - (i i) l^{2} + m^{2} - n^{2} = 0 - (i i i) (i i) + (i i i) 2 (l^{2} + m^{2}) = 1 2 n^{2} = 1 \Rightarrow n = \pm \frac{1}{\sqrt{2}} l^{2} + m^{2} = \frac{1}{2} l + m = \frac{1}{\sqrt{2}} l^{2} + {(\frac{1}{\sqrt{2}} - l)}^{2} = \frac{1}{2} l^{2} + l^{2} + \frac{1}{2} - \sqrt{2} l = \frac{1}{2} 2 l^{2} - \sqrt{2} l = 0 l = 0, \frac{1}{\sqrt{2}} \Rightarrow m = \frac{1}{\sqrt{2}}, 0 l^{2} + m^{2} = \frac{1}{2} l + m = \frac{- 1}{\sqrt{2}} l^{2} + {(\frac{- 1}{\sqrt{2}} - l)}^{2} = \frac{1}{2} l^{2} + \frac{1}{2} + l^{2} + \sqrt{2} l = \frac{1}{2} 2 l^{2} + \sqrt{2} l = 0 l = 0, \frac{- 1}{\sqrt{2}} \Rightarrow \frac{- 1}{\sqrt{2}}, 0$
Possible values of $(l, m, n)$ are:
$(0, \frac{1}{\sqrt{2}}, \frac{- 1}{\sqrt{2}}), (\frac{1}{\sqrt{2}}, 0, \frac{- 1}{\sqrt{2}}), (0, \frac{- 1}{\sqrt{2}}, \frac{1}{\sqrt{2}}), (\frac{- 1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}})$

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