Question

A line $AB$ in three-dimensional space males angles ${45}^o$ and ${120}^o$ with the positive x-axis and the positive y-axis respectively. If $AB$ makes an acute angle $\theta$ with the positive z-axis, then $\theta$ equal

• A. ${45}^o$
• B. ${60}^o$
• C. ${75}^o$
• D. ${30}^o$

Answer 1

Answer: (B)

${60}^o$

'l', 'm', 'n' be the Direction cosine of the line

$l=cos\alpha =cos{45}^{\circ }=\frac{1}{\sqrt{2}}$

$m=cos\beta =cos{120}^{\circ }=\frac{-1}{2}$

$n=cos\theta$

we know

$l^2+m^2+n^2=1$

$\Rightarrow d\frac{1}{2}+\frac{1}{4}+co{s}^2\theta =1$

$\Rightarrow co{s}^2\theta =1-\frac{1}{2}-\frac{1}{4}$

$\Rightarrow co{s}^2\theta =\frac{1}{4}$

$\Rightarrow cos=\frac{1}{2}$

$\therefore \theta ={60}^{\circ }$