If a line makes angles $\frac{\pi}{4}$ and $\frac{\pi}{3}$ with $Y$-axis and $Z$-axis respectively, then the…
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
$\frac{\pi}{3}$
$\frac{2 \pi}{3}$
$\frac{\pi}{6}$
$\frac{5 \pi}{6}$
Solution
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
$
$
\begin{gathered}
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}
\end{gathered}
$
$\Rightarrow \cos \alpha=-\frac{1}{2}(\alpha$ is obtuse angle $)$
$
\Rightarrow \alpha=\pi-\pi / 3=2 \pi / 3 .
$