If two lines, whose equations are $\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-1}{\lambda }$ and $\frac{x-2}{3}=\frac{y-3}{3}=\frac{z-2}{3}$ lie in the same plane. Then, the value of ${\sin }^{-1}\left(\sin \lambda \right)$ is

$\text{ These lines lie in the same plane. So, both lines are coplanar. }$

Coplanar condition is det is zero.

$\therefore \left|\begin{array}{ccc}3-2& 2-3& \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1-2\\ 2& 3& \lambda \\ 3& 3& 3\end{array}\right|=0$

$\lambda =3$

${\sin }^{-1}\sin \lambda ={\sin }^{-1}\sin \left(3\right)=\sin (\pi -\theta )=\sin \theta$

$\sin (\pi -3)=\sin 3$

${\sin }^{-1}\sin (\pi -3)=4^{th}quardant$

${\sin }^{-1}\sin \theta =\theta \Rightarrow \theta liesbetween-\frac{\pi }{2}\le \theta \le \frac{\pi }{2}$

${\sin }^{-1}\sin (\pi -3)=\pi -3$