Question

The plane $2x-3y+6z-11=0$ makes an angle $si{n}^{-1}\left(\alpha \right)$ with X-axis. The value of $\alpha$ is equal to

• A. $\frac{2}{7}$
• B. $\frac{3}{7}$
• C. $\frac{\sqrt{2}}{7}$
• D. $\frac{\sqrt{3}}{7}$

Answer 1

The correct option is A $\frac{2}{7}$

Equation of plane is $2x-3y+6z-11=0$
Or $\overrightarrow{n}=2\hat{i}-3\hat{j}+6\hat{k}$
Equation of line is $\hat{b}=\hat{i}$
$\sin \theta =\frac{\overrightarrow{b}\cdot \overrightarrow{n}}{\left|\overrightarrow{b}\right|\left|\overrightarrow{n}\right|}$
$\sin \left({\sin }^{-1}\alpha \right)=\frac{(2\hat{i}-3\hat{j}+6\hat{k})\cdot (\hat{i}+0\hat{j}+0\hat{k})}{\sqrt{{\left(2\right)}^2+{(-3)}^2+{\left(6\right)}^2}\cdot \sqrt{{\left(1\right)}^2}}$
$\alpha =\frac{2}{\sqrt{49}}$
$\alpha =\frac{2}{7}$