Question

If the plane $2x-3y-6z=13$ makes an angle ${\sin }^{-1}\left(\lambda \right)$ with the x-axis, then find the value of $\lambda$.

Answer 1

Direction ratios of the x-axis are $1,0,0$ and direction ratios of normal to the plane are $2,-3,-6$.

Now, the angle between the line and the plane is given by:

$\sin \varphi =\frac{|\overrightarrow{a}.\overrightarrow{n}|}{|\overrightarrow{a}||\overrightarrow{n}|}$

$\sin \varphi =\frac{|a_1.a_2+b_1.b_2+c_1.c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}.\sqrt{a_2^2+b_2^2+c_2^2}}$

Let $\varphi$ be the angle between the x-axis and the given plane.

Then,

$\sin \varphi =\frac{|1\times 2+0\times (-3)+0\times (-6)|}{\left\{\sqrt{1^2+0^2+0^2}\right\}\left\{\sqrt{2^2+(-3{)}^2+(-6{)}^2}\right\}}=\frac{2}{7}$

$\Rightarrow \varphi ={\sin }^{-1}\left(\frac{2}{7}\right)$.

Hence, $\lambda =\frac{2}{7}$.