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  4. The number of lines making equal angles with the coordinate axes in three dimensional geometry is equal to

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Q. The number of lines making equal angles with the coordinate axes in three dimensional geometry is equal to

J & K CETJ & K CET 2011Three Dimensional Geometry

Solution:

Let l, m and n be the direction cosines of a line.
Since, the line is equally inclined with OX, OY and OZ.
$ \therefore $ $ l=m-n $ $ (\because \,\,\,\cos \,\,\alpha =\cos \beta =\cos \gamma ) $
Now, $ {{l}^{2}}+{{m}^{2}}+{{n}^{2}}=1\Rightarrow 3{{l}^{2}}=1 $
$ \Rightarrow $ $ {{l}^{2}}=1/3 $
$ \Rightarrow $ $ l=\pm \frac{1}{\sqrt{3}} $
Hence, the direction cosines of given line are
$ \pm \frac{1}{\sqrt{3}},\,\,\pm \frac{1}{\sqrt{3}},\,\,\pm \,\frac{1}{\sqrt{3}}, $
Since, $ +ve $ and $ -ve $
signs can be arranged at three places in
$ 2\times 2\times 2=8 $ ways.
Therefore, there are eight lines which are equally inclined with the coordinate axis.