Question

What are the direction cosines of a line which makes equal angles with the coordinate axes?

Answer 1

First of all, we take the line and assume the angles with the axes. The equal angles give as the equal value of the 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 the direction cosines of the given lines which make equal angles with the given coordinate axes.

Complete step by step answer:
From the given, that 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 angle to be $\alpha = \beta = \gamma = \theta$.
The direction cosines of the angles will be $\cos \alpha ,\cos \beta ,\cos \gamma$.
Since we know the property for the given direction cosines of the angles as ${\cos ^2}\alpha + {\cos ^2}\beta + {\cos ^2}\gamma = 1$
Now putting the values as we know, ${\cos ^2}\alpha + {\cos ^2}\beta + {\cos ^2}\gamma = 1 \Rightarrow {\cos ^2}\theta + {\cos ^2}\theta + {\cos ^2}\theta = 1$ (because all are the same)
Thus, solving we get $3{\cos ^2}\theta = 1 \Rightarrow {\cos ^2}\theta = \dfrac{1}{3}$
Hence, we get after taking the square root terms, ${\cos ^2}\theta = \dfrac{1}{3} \Rightarrow \cos \theta = \pm \dfrac{1}{{\sqrt 3 }}$
Therefore, the direction of the cosines is represented as $\left( {\cos \alpha ,\cos \beta ,\cos \gamma } \right) = \left( { \pm \dfrac{1}{{\sqrt 3 }}, \pm \dfrac{1}{{\sqrt 3 }}, \pm \dfrac{1}{{\sqrt 3 }}} \right)$
We are also able to find its ratio which is any number proportional to the direction cosine is known as the direction ratio of the line. Here the direction cosines are equal and hence we get the direction ratio as $\cos \alpha :\cos \beta :\cos \gamma = 1:1:1$

Note:
The direction cosines of the line parallel to any coordinate axis are equal to the direction cosines of the coordinate axis. They are also denoted as $\left( {\cos \alpha ,\cos \beta ,\cos \gamma } \right) = \left( {l,m,n} \right)$ and then we get ${l^2} + {m^2} + {n^2} = 1$ and thus we may also able to use this method, but we only get the same answer as the directions are $\left( {\cos \alpha ,\cos \beta ,\cos \gamma } \right) = \left( { \pm \dfrac{1}{{\sqrt 3 }}, \pm \dfrac{1}{{\sqrt 3 }}, \pm \dfrac{1}{{\sqrt 3 }}} \right)$ and the ratio is $\cos \alpha :\cos \beta :\cos \gamma = 1:1:1$