Question

Solve!!

please give a detailed solution! ​

Answer 1

$\large\underline{\sf{Solution-}}$

Let assume that the line makes an angle α, β, γ with the respective x, y and z axis respectively.

According to statement,

It is given that

$\rm :\longmapsto\: \gamma = 60 \degree$

and

$\rm :\longmapsto\:cos \beta \: : \: cos \alpha = \sqrt{3} \: : \: 1$

So,

$\rm :\longmapsto\:cos \beta = \sqrt{3}k$

and

$\rm :\longmapsto\:cos \alpha = k$

We know that,

$\rm :\longmapsto\: {cos}^{2} \alpha + {cos}^{2} \beta + {cos}^{2} \gamma = 1$

$\rm :\longmapsto\: {k}^{2} + 3 {k}^{2} + {\bigg(\dfrac{1}{2} \bigg) }^{2} = 1$

$\rm :\longmapsto\:4{k}^{2} = 1 - \dfrac{1}{4}$

$\rm :\longmapsto\:4{k}^{2} = \dfrac{4 - 1}{4}$

$\rm :\longmapsto\:4{k}^{2} = \dfrac{3}{4}$

$\rm :\longmapsto\:{k}^{2} = \dfrac{3}{16}$

$\rm :\longmapsto\:{k} \: = \: \pm \: \dfrac{ \sqrt{3} }{4}$

So, we get 2 cases arises for the position of the line.

Case :- 1 When

$\rm :\longmapsto\:{k} \: = \: \dfrac{ \sqrt{3} }{4}$

So,

$\rm :\longmapsto\:cos \alpha = \dfrac{ \sqrt{3} }{4}$

$\rm :\longmapsto\:cos \beta = \dfrac{3}{4}$

$\rm :\longmapsto\:cos \gamma = \dfrac{1}{2}$

So, Direction cosines of the line is

$\rm :\longmapsto\:\bigg(\dfrac{ \sqrt{3} }{4}, \: \dfrac{3}{4}, \: \dfrac{1}{2} \bigg)$

Case :- 2 When

$\rm :\longmapsto\:{k} \: = \: - \: \dfrac{ \sqrt{3} }{4}$

So,

$\rm :\longmapsto\:cos \alpha = - \dfrac{ \sqrt{3} }{4}$

$\rm :\longmapsto\:cos \beta = - \: \dfrac{3}{4}$

$\rm :\longmapsto\:cos \gamma = \dfrac{1}{2}$

So, Direction cosines of line is

$\rm :\longmapsto\:\bigg(\dfrac{ - \: \sqrt{3} }{4}, \: - \: \dfrac{3}{4}, \: \dfrac{1}{2} \bigg)$

Now,

Let suppose that the angle between the two position of lines be 'p'.

So, we have Direction cosines of lines in two different positions as

$\rm :\longmapsto\:\bigg(\dfrac{ \: \sqrt{3} }{4}, \: \: \dfrac{3}{4}, \: \dfrac{1}{2} \bigg)$

and

$\rm :\longmapsto\:\bigg(\dfrac{ - \: \sqrt{3} }{4}, \: - \: \dfrac{3}{4}, \: \dfrac{1}{2} \bigg)$

Thus,

Angle between the lines is

$\rm :\longmapsto\:cosp \: = |\dfrac{ - \sqrt{3} }{4} \times \dfrac{ \sqrt{3} }{4} + \dfrac{ - 3}{4} \times \dfrac{3}{4} + \dfrac{1}{2} \times \dfrac{1}{2} |$

$\rm \: = \: | - \dfrac{3}{16} - \dfrac{9}{16} + \dfrac{1}{4} |$

$\rm \: = \: | - \dfrac{3}{16} - \dfrac{9}{16} + \dfrac{4}{16} |$

$\rm \: = \: | - \dfrac{8}{16} |$

$\rm \: = \: | - \dfrac{1}{2} |$

$\rm \: = \: \dfrac{1}{2}$

$\bf\implies \:p \: = \: 60 \degree$