The angle between the line $\frac{x-1}{2}=\frac{y+3}{1}=\frac{z+7}{2}$ and the plane $\bar{r} \cdot(6 \hat{\imath}-2 \hat{\jmath}-3 \hat{k})=5$ (A) $\sin ^{-1}\left(\frac{4}{21}\right)$ (B) $\cos ^{-1}\left(\frac{4}{21}\right)$ (C) $\sin ^{-1}\left(\frac{5}{7}\right)$ (D) $\operatorname{COS}^{-1}\left(\frac{5}{7}\right)$
The angle between the line $\frac{x-1}{2}=\frac{y+3}{1}=\frac{z+7}{2}$ and the plane $\bar{r} \cdot(6 \hat{\imath}-2 \hat{\jmath}-3 \hat{k})=5$ (A) $\sin ^{-1}\left(\frac{4}{21}\right)$ (B) $\cos ^{-1}\left(\frac{4}{21}\right)$ (C) $\sin ^{-1}\left(\frac{5}{7}\right)$ (D) $\operatorname{COS}^{-1}\left(\frac{5}{7}\right)$

The correct option is option(A)

we know the angle between the line. $\frac{x-x_{0}}{l}=\frac{y-y_{0}}{m}=\frac{z-z_{0}}{n}$ and the plane $A x+B y +Cz+D=0$ is given $\psi=\sin ^{-1} \frac{A l+B m+C n}{\sqrt{A^{2}+B^{2}+c^{2}} \cdot \sqrt{l^{2}+m^{2}+n^{2}}}=$
equation of plane in paramterized form $\Rightarrow 6 x-2 y-3 k-5=0$
=> $l=2, m=1 \quad n=2 \quad A=6 \\B=-2 \quad c=-3$ $D=-5$
using thse values
$\begin{array}{l}
\psi=\sin ^{-1} \frac{(6 \times 2)+(-2 \times 1)+(-3 \times 2)}{\sqrt{(6)^{2}+(-2)^{2}+(-3)^{2}} \sqrt{(2)^{2}+(1)^{2}+(2)^{2}}} \\
\psi=\sin ^{-1} \frac{12-2-6}{\sqrt{36+4+3} \sqrt{4+1+4}} \\
\psi=\sin ^{-1} \frac{4}{7 \times 3}=> \sin ^{-1}\left(\frac{4}{21}\right)
\end{array}$