Let L be the line of intersection of the planes \(2 x+3 y+z=1\) and \(x+3 y+2 z=2\). If L makes an angle \(\alpha\) with the positive X -axis, then \(\cos \alpha\) equals

\(\mathrm{n}_1=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\hat{\mathrm{k}} \text { and } \overline{\mathrm{n}}_2=\hat{\mathrm{i}}+3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}\)
\(\therefore \quad\) The line L is parallel to
\(\begin{aligned}
& \overline{\mathrm{n}}=\overline{\mathrm{n}}_1 \times \overline{\mathrm{n}}_2=\left|\begin{array}{ccc}
\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\
2 & 3 & 1 \\
1 & 3 & 2
\end{array}\right|=3 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+3 \hat{\mathrm{k}} \\
& \Rightarrow \cos \alpha=\frac{\overline{\mathrm{n}} \cdot \hat{\mathrm{i}}}{|\overline{\mathrm{n}}||\hat{\mathrm{i}}|}=\frac{3}{3 \sqrt{3}}=\frac{1}{\sqrt{3}}
\end{aligned}\)