If a line in space makes equal angles with the coordinate axes, then find the direction cosines and direction ratios of this line.

Hint: We first take the line and assume the angles with the axes. The equal angles give equal value of cosine which creates the trigonometric equation as ${{\cos }^{2}}\theta =\dfrac{1}{3}$ from the property of ${{\cos }^{2}}\alpha +{{\cos }^{2}}\beta +{{\cos }^{2}}\gamma =1$. We solve it to find both direction cosines and direction ratios as direction ratio is proportional to direction cosines.

Complete step by step solution:
It is given that a line in space makes equal angles with the coordinate axes. We first take the line and assume the angles with the axes.
We assume the line L makes angles $\alpha ,\beta ,\gamma $ with the X, Y, Z axes respectively.
As the angles are all equal, we can assume that the angles to be $\alpha =\beta =\gamma =\theta $.
The direction cosines of the angles will be $\cos \alpha ,\cos \beta ,\cos \gamma $.
We know the property for the direction cosines of the angles as ${{\cos }^{2}}\alpha +{{\cos }^{2}}\beta +{{\cos }^{2}}\gamma =1$.
Putting the values, we get ${{\cos }^{2}}\theta +{{\cos }^{2}}\theta +{{\cos }^{2}}\theta =3{{\cos }^{2}}\theta =1$.
We now simplify the equation to get ${{\cos }^{2}}\theta =\dfrac{1}{3}$ which gives $\cos \theta =\pm \dfrac{1}{\sqrt{3}}$.
Therefore, the direction cosines are $\left( \cos \alpha ,\cos \beta ,\cos \gamma \right)=\left( \pm \dfrac{1}{\sqrt{3}},\pm \dfrac{1}{\sqrt{3}},\pm \dfrac{1}{\sqrt{3}} \right)$.
For direction ratio, we know that any number proportional to the direction cosine is known as the direction ratio of a line. Here the direction cosines are equal and therefore, the direction ratio is $1:1:1$.

Note: The direction cosines of a line parallel to any coordinate axis are equal to the direction cosines of the corresponding axis. They are also denoted as $\left( \cos \alpha ,\cos \beta ,\cos \gamma \right)=\left( l,m,n \right)$ and we get ${{l}^{2}}+{{m}^{2}}+{{n}^{2}}=1$.