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If the direction cosines $(1, \mathrm{~m}, \mathrm{n})$ of two lines are connected $\mathrm{b}$ the relations $1+\mathrm{m}+\mathrm{n}=0$ and $\mathrm{lm}=0$, then the angle between those lines is
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$\frac {\pi}{3}$
$l+m+n=0 ; l m=0$
$\begin{array}{lll}\Rightarrow & l=0 \text { or } m=0 & \\ & \text { when } l=0 & \text { when } m=0 \\ \Rightarrow & m+n=0 & l+n=0 \\ \Rightarrow & n=-m & n=-l \\ \therefore \quad & \text { D.R. }=(0, m,-m) \quad \therefore \text { DR }(l, 0,-l) \\ & (0,1,-1) \text { and }(1,0,-1) \\ \therefore \quad & \cos \theta=\frac{1(0)+1(0)+(-1)(-1)}{\sqrt{0^2+1^2+(-1)^2} \cdot \sqrt{1^2+0^2+(-1)^2}} \\ & =\frac{1}{\sqrt{2} \cdot \sqrt{2}}=\frac{1}{2} \therefore \quad \theta=\frac{\pi}{3} .\end{array}$
$\begin{array}{lll}\Rightarrow & l=0 \text { or } m=0 & \\ & \text { when } l=0 & \text { when } m=0 \\ \Rightarrow & m+n=0 & l+n=0 \\ \Rightarrow & n=-m & n=-l \\ \therefore \quad & \text { D.R. }=(0, m,-m) \quad \therefore \text { DR }(l, 0,-l) \\ & (0,1,-1) \text { and }(1,0,-1) \\ \therefore \quad & \cos \theta=\frac{1(0)+1(0)+(-1)(-1)}{\sqrt{0^2+1^2+(-1)^2} \cdot \sqrt{1^2+0^2+(-1)^2}} \\ & =\frac{1}{\sqrt{2} \cdot \sqrt{2}}=\frac{1}{2} \therefore \quad \theta=\frac{\pi}{3} .\end{array}$
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