If a line L makes angles $\frac{\pi}{3}$ and $\frac{\pi}{4}$ with Y -axis and Z -axis respectively, then the…
If a line L makes angles $\frac{\pi}{3}$ and $\frac{\pi}{4}$ with Y -axis and Z -axis respectively, then the angle between L and another line having direction ratios $1,1,1$ is
L makes $\frac{\pi}{3}$ and $\frac{\pi}{4}$ angle with Y -axis and Z -axis
$\therefore m=\cos \frac{\pi}{3}=\frac{1}{2}$ and $n=\cos \frac{\pi}{4}=\frac{1}{\sqrt{2}}$
and $l^2+m^2+n^2=1 \Rightarrow l^2+\frac{1}{4}+\frac{1}{2}=1 \Rightarrow l=\frac{1}{2}$
Now, angle between $L$ and line with direction ratio's $ \lt 1,1,1\gt$ is
$\begin{aligned} & \cos \theta=\left(\frac{1 \times \frac{1}{2}+1 \times \frac{1}{2}+1 \times \frac{1}{\sqrt{2}}}{\sqrt{\frac{1}{4}+\frac{1}{4}+\frac{1}{2}} \sqrt{1+1+1}}\right)=\frac{\sqrt{2}+1}{\sqrt{6}} \\ & \Rightarrow \theta=\cos ^{-1}\left(\frac{\sqrt{2}+1}{\sqrt{6}}\right)\end{aligned}$