Question
The plane $$2x - 3y + 6z - 11 = 0$$ makes an angle $${\sin ^{ - 1}}\left( a \right)$$ with the $$x$$-axis. Then the value of $$a$$ is –
A.
$$\frac{{\sqrt 3 }}{2}$$
B.
$$\frac{{\sqrt 2 }}{3}$$
C.
$$\frac{3}{7}$$
D.
$$\frac{2}{7}$$
Answer :
$$\frac{2}{7}$$
Solution :
$$\because $$ DR's of the normal of the plane $$ = 2, - 3, 6$$
and DR's of $$x$$-axis is $$1,\,0,\,0$$
and If $$\theta $$ is the angle between the plane
and $$x$$-axis, then
$$\eqalign{
& \sin \,\theta = \frac{{2\left( 1 \right) + \left( { - 3} \right)\left( 0 \right) + \left( 6 \right)\left( 0 \right)}}{{\sqrt {{2^2} + {{\left( { - 3} \right)}^2} + {6^2}} \,\sqrt 1 }} \cr
& \Rightarrow \sin \,\theta = \frac{2}{7} \cr
& \Rightarrow \theta = {\sin ^{ - 1}}\left( {\frac{2}{7}} \right) \cr
& \therefore \,a = \frac{2}{7} \cr} $$