Question

How many straight lines are there that are equally aligned to three-dimensional coordinate axes?

Answer 1

Hint:Direction cosines of a vector are the ones that determine its direction in space. They are the cosines of the angle that the vector makes with the three-dimensional coordinate axes, respectively. If $(l,m,n)$ are the direction cosines of a line in space, then ${l^2} + {m^2} + {n^2} = 1$ .

Complete step by step Solution:
Let $\alpha ,\beta ,\gamma $ be the angles that a straight line makes with the three-dimensional coordinate axes, respectively.
Then, the direction cosines of the line are:
$l = \cos \alpha $ ,
$m = \cos \beta $ , and
$n = \cos \gamma $
Now, as $(l,m,n)$ are the direction cosines of the line, therefore,
${l^2} + {m^2} + {n^2} = 1$
Substituting their values,
${\cos ^2}\alpha + {\cos ^2}\beta + {\cos ^2}\gamma = 1$ … (1)
As it is given that the line is equally aligned with all the three-dimensional coordinate axes, hence,
$\alpha = \beta = \gamma $
This also implies that $\cos \alpha = \cos \beta = \cos \gamma $ .
Substituting this in equation (1),
$3{\cos ^2}\alpha = 1$
Simplifying further and taking the square root, we have:
$\cos \alpha = \pm \dfrac{1}{{\sqrt 3 }}$
From the above equation, we can induce all the combinations of the direction cosines possible, hence, giving us all the lines possible.
Thus, all the combinations are:
$\left( {\dfrac{1}{{\sqrt 3 }},\dfrac{1}{{\sqrt 3 }},\dfrac{1}{{\sqrt 3 }}} \right),\left( {\dfrac{1}{{\sqrt 3 }},\dfrac{1}{{\sqrt 3 }}, - \dfrac{1}{{\sqrt 3 }}} \right),\left( {\dfrac{1}{{\sqrt 3 }}, - \dfrac{1}{{\sqrt 3 }},\dfrac{1}{{\sqrt 3 }}} \right),\left( { - \dfrac{1}{{\sqrt 3 }},\dfrac{1}{{\sqrt 3 }},\dfrac{1}{{\sqrt 3 }}} \right)$
Thus, there are a total of four straight lines that can be equally aligned with all the three-dimensional axes.

Note:Using permutations, the possible number of combinations that can be formed is 8. However, in the above question, it is important to note that half of the combinations, of the direction ratios, obtained will result in the formation of the same line.