Question

Answer this Question of iit jee . Chapter → Three dimensional Geometry ​

Answer 1

Answer:

Line AB makes angle 45° with positive x-axis,,

$→ direction \: cosine , l = cos 45° = \frac{1}{ \sqrt{2} }$

Line AB makes angle 120° with positive y-axis,,

$→ direction \: cosine, \: m = cos120° = \frac{ - 1}{2}$

Lime AB makes angle θ with positive z-axis,,

→ direction cosine, n = cosθ

Now,

l² + m² + n² = 1

$→ { (\frac{1}{ \sqrt{2} } })^{2} + {( \frac{ - 1}{2} })^{2} + cosθ = 1$

$→\frac{1}{2} + \frac{1}{4} + { \cos }^{2} θ = 1$

$→{ \cos }^{2}θ = 1 - \frac{1}{2} - \frac{1}{4}$

$→ { \cos }^{2}θ = \frac{4 - 2 - 1}{4}$

$→ { \cos }^{2}θ = \frac{1}{4}$

$→cosθ = ± \frac{1}{ 2}$

Since, θ is an acute angle .,,

$∴cosθ = \frac{1}{2 }$

$∴θ = 60°$

Answer 2

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

Let assume that direction cosines of line AB be ( l, m, n).

We know,

Direction cosines of the line is defined as cosine of the angle which a line makes with x - axis, y - axis and z - axis respectively.

$\rm :\longmapsto\:If \: a \: line \: makes \: an \: angle \: \alpha , \beta , \gamma \: withx,y,z \: axis,\: then$

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

$\rm :\longmapsto\:m = cos \beta$

$\rm :\longmapsto\:n = cos \gamma$

According to statement,

It is given that line AB makes an angle of 45° with x - axis.

So,

$\rm :\longmapsto\:l = cos 45 \degree = \dfrac{1}{ \sqrt{2} }$

Also, it is given that line AB makes an angle of 120° with y - axis respectively,

So,

$\rm \longmapsto\:m= cos120\degree =cos(180\degree- 60\degree) = - cos60\degree=- \dfrac{1}{ 2}$

Also, given that line AB makes an acute angle θ with z - axis.

So,

$\rm :\longmapsto\:n = cos \theta$

Now,

We know that,

$\rm :\longmapsto\: {l}^{2} + {m}^{2} + {n}^{2} = 1$

On substituting the values of l, m and n, then

$\rm :\longmapsto\:\dfrac{1}{2} + \dfrac{1}{4} + {cos}^{2}\theta = 1$

$\rm :\longmapsto\:\dfrac{2 + 1}{4} + {cos}^{2}\theta = 1$

$\rm :\longmapsto\:\dfrac{3}{4} + {cos}^{2}\theta = 1$

$\rm :\longmapsto\: {cos}^{2}\theta = 1 - \dfrac{3}{4}$

$\rm :\longmapsto\: {cos}^{2}\theta = \dfrac{4 - 3}{4}$

$\rm :\longmapsto\: {cos}^{2}\theta = \dfrac{1}{4}$

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

$\rm :\longmapsto\:As \: it \: is \: given \: that \: \theta \: is \: acute$

Therefore,

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

$\rm :\longmapsto\: {cos}\theta \: = \: cos60\degree$

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

Additional Information :-

$\rm :\longmapsto\:If \: a \: line \: makes \: an \: angle \: \alpha , \beta , \gamma \: withx,y,z \: axis,\: then$

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

$\rm :\longmapsto\:(2). \: \: {sin}^{2} \alpha + {sin}^{2} \beta + {sin}^{2} \gamma = 2$