Line and Plane — Maths STD 12 Science — Question

$\sin \theta=\left|\frac{\bar{b} \cdot \bar{n}}{|b \vec{b}||n|}\right|$

$\ldots(1)$

$\begin{aligned} \text { Here, } \bar{b} & =2 \hat{i}+3 \hat{j}-6 \hat{k}, \bar{n}=2 \hat{i}-\hat{j}+\hat{k} \\ \therefore \bar{b} \cdot \bar{n} & =(2 \hat{i}+3 \hat{j}-6 \hat{k}) \cdot(2 \hat{i}-\hat{j}+\hat{k}) \\ & =(2)(2)+(3)(-1)+(-6)(1) \\ & =4-3-6=-5\end{aligned}$

Also, $|\bar{b}|=\sqrt{2^2+3^2+(-6)^2}=\sqrt{49}=7$

$|\bar{n}|=\sqrt{2^2+(-1)^2+1^2}=\sqrt{6}$

$\therefore$ from $(1)$, we have

$\begin{aligned} & \sin \theta=\left|\frac{-5}{7 \sqrt{6}}\right|=\frac{5}{7 \sqrt{6}} \\ & \therefore \theta=\sin ^{-1}\left(\frac{5}{7 \sqrt{6}}\right) .\end{aligned}$