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  4. The number of straight lines that are equally inclined to the three-dimensional coordinate axes, is

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Q. The number of straight lines that are equally inclined to the three-dimensional coordinate axes, is

Jharkhand CECEJharkhand CECE 2013

Solution:

Since, $ \alpha =\beta =\gamma $
$\Rightarrow {{\cos }^{2}}\alpha +{{\cos }^{2}}\alpha +{{\cos }^{2}}\alpha =1 $
$ \Rightarrow $ $ \Rightarrow \alpha +{{\cos }^{-}}\left( \pm \frac{1}{\sqrt{3}} \right) $
So, there are four line whose $ DC's $ are $ \left( \frac{1}{\sqrt{3}},\,\,\frac{1}{\sqrt{3}},\,\,\frac{1}{\sqrt{3}} \right),\,\,\left( \frac{-1}{\sqrt{3}},\,\,\frac{1}{\sqrt{3}},\,\,\frac{1}{\sqrt{3}} \right),\,\,\left( \frac{1}{\sqrt{3}},\,\,\frac{-1}{\sqrt{3}},\,\,\frac{1}{\sqrt{3}} \right) $
$ \left( \frac{1}{\sqrt{3}},\,\,\frac{1}{\sqrt{3}},\,\,\frac{-1}{\sqrt{3}} \right) $