A line L is parallel to both the planes$2x+3y+z=1$ and $x+3y+2z=2.$ If line L makes an angle $\alpha$ with the positive direction of X-axis, then $\cos \alpha =$

Let the direction cosines of line L be I, m and n. Since. the line intersect the given planes, then the normal to the planes are perpendicular to the line L

$\therefore 2l+3m+n=0$

and $l+3m+2n=0$

From Eqs. (i) and (ii), we get

$\frac{l}{3}=\frac{m}{-3}=\frac{n}{3}=k$

We know that $l^2+m^2+n^2=1$

$\begin{array}{l}\therefore (3k{)}^2+(-3k{)}^2+(3k{)}^2=1\\ \Rightarrow 27k^2=1\\ \Rightarrow k=\frac{1}{3\sqrt{3}}\\ \therefore l=\frac{1}{\sqrt{3}}\\ \cos \alpha =\frac{1}{\sqrt{3}}\end{array}$