The plane 2x - 3y + 6z - 11 = 0 makes an angle sin^-1 (∝) with X-axis . the value of ∝ is equal to

Concept:

If the equation of the line is an \(\rm \vec r = \;\vec a + \lambda \;\vec b\) and equation of the plane is \(\rm \overrightarrow{r}. \overrightarrow{n}=d\) . then the angle α  between line and direction parallel to the plane is, 

\(\rm \sin α = \left | \frac{\overrightarrow{b}.\overrightarrow{n}}{\left | \overrightarrow{b} \right |\left | \overrightarrow{n} \right |} \right |\)  

Calculation: 

Equation of line with X-axis is,

\(\rm \overrightarrow{b}= \hat{i}\)

Equation of plane is  2x -3y +6z - = 0 

or in  vector form,  \(\rm \overrightarrow{n}= 2\hat{i}-3\hat{j}+6\hat{k}\)   

∴   \(\rm \sin α = \left | \frac{\overrightarrow{b}.\overrightarrow{n}}{\left | \overrightarrow{b} \right |\left | \overrightarrow{n} \right |} \right |\)

⇒ \(\rm \sin α = \frac{\left ( 2\hat{i}-3\hat{j}+6\hat{k} \right ). \left ( \hat{i}+ 0\hat{j}+0\hat{k} \right )}{\sqrt{2^{2}+(-3)^{2}+6^{2}}. \sqrt{1^{2}}}\) = \(\frac{2}{\sqrt{49}}\)

⇒ sin α = \(\frac{2}{7}\) 

⇒ α = sin^-1 ( \(\frac{2}{7}\) )  

⇒ α = \(\frac{2}{7}\).

The correct option is 2.