Answer:
Let the line makes an angle a with each of the three coordinate axes, then its direction cosines are \[l=\cos \alpha ,\]\[m=\cos \alpha \]and \[n=\cos \alpha \]. We know that, \[{{l}^{2}}+{{m}^{2}}+{{n}^{2}}=1\] \[\Rightarrow \]\[{{\cos }^{2}}\alpha +{{\cos }^{2}}\alpha +{{\cos }^{2}}\alpha =1\]\[\Rightarrow \]\[3{{\cos }^{2}}\alpha =1\] \[\Rightarrow \]\[{{\cos }^{2}}\alpha =\frac{1}{3}\Rightarrow \cos \alpha =\pm \,\,\frac{1}{\sqrt{3}}\] Hence, the direction cosines of a line are either \[\frac{1}{\sqrt{3}},\,\,\frac{1}{\sqrt{3}},\,\,\frac{1}{\sqrt{3}}\]or\[\frac{-\,1}{\sqrt{3}},\,\,\frac{-\,1}{\sqrt{3}},\,\,\frac{-\,1}{\sqrt{3}}\].
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