If the line of intersection of the planes $a x+b y=3$ and $a x+b y+c z=0$, a $>0$ makes an angle $30^{\circ}$ with the plane $y-z+2=0$, then the direction cosines of the line are :

<p>${P_1}:ax + by + 0z = 3$, normal vector : ${\overrightarrow n _1} = (a,b,0)$</p> <p>${P_2}:ax + by + cz = 0$, normal vector : ${\overrightarrow n _2} = (a,b,c)$</p> <p>Vector parallel to the line of intersection $= {\overrightarrow n _1} \times {\overrightarrow n _2}$</p> <p>${\overrightarrow n _1} \times {\overrightarrow n _2} = (bc, - ac,0)$</p> <p>Vector normal to $0\,.\,x + y - z + 2 = 0$ is ${\overrightarrow n _3} = (0,1, - 1)$</p> <p>Angle between line and plane is 30$^\circ$</p> <p>$$ \Rightarrow \left| {{{0 - ac + 0} \over {\sqrt {{b^2}{c^2} + {c^2}{a^2}} \sqrt 2 }}} \right| = {1 \over 2}$$</p> <p>$\Rightarrow {a^2} = {b^2}$</p> <p>Hence, ${\overrightarrow n _1} \times {\overrightarrow n _2} = (ac, - ac,0)$</p> <p>Direction ratios $= \left( {{1 \over {\sqrt 2 }}, - {1 \over {\sqrt 2 }},0} \right)$</p>