Question

If a line makes angles $\frac{\pi }{4}$ and $\frac{\pi }{3}$ with $Y$ -axis and $Z$ -axis respectively, then the obtuse angle made by that line with $X$ - axis is

• A. $\frac{\pi }{3}$
• B. $\frac{2\pi }{3}$
• C. $\frac{\pi }{6}$
• D. $\frac{5\pi }{6}$

Answer 1

Answer: (B)

Let line makes angle $\alpha$ with $X$ -axis then, direction cosine of the line are
$l=\cos \alpha ,m=\cos \frac{\pi }{4},n=\cos \frac{\pi }{3}$
But $l^2+m^2+n^2=1$
$l^2+{\left(\frac{1}{\sqrt{2}}\right)}^2+{\left(\frac{1}{2}\right)}^2=1$
$l^2+\frac{1}{2}+\frac{1}{4}=1$
$l^2=\frac{1}{4}$
$\Rightarrow l=\pm \frac{1}{2}$
$\Rightarrow \cos \alpha =-\frac{1}{2}(\alpha$ is obtuse angle $)$
$\Rightarrow \alpha =\pi -\pi /3=2\pi /3$