The angle between the lines x + 2y +3z = 0 = 3x +2y +z and the y – axis is

$$\begin{gathered} Let \left(l,m,n \right) be the drs of the line \\ \Rightarrow \mathcal{l}+2\mathrm{m}+3\mathrm{n}=0, 3\mathcal{l}+2\mathrm{m}+\mathrm{n}=0 \\ solving both equations \\ \Rightarrow (\mathcal{l},\mathrm{m},\mathrm{n})=(1,-2,1) \\ dcs of \mathrm{y}-\mathrm{axis} =(0,1,0) \\ \theta be the angle between y axis and given line \\ \Rightarrow \cos \mathrm{\theta }=\frac{-2}{\sqrt{6}\sqrt{1}}=\frac{-2}{\sqrt{6}} \\ \therefore \cos 2\mathrm{\theta }=2{\cos }^2 \mathrm{\theta }-1=2\times \frac{4}{6}-1=\frac{4}{3}-1=\frac{1}{3} \\ \Rightarrow \sec 2\mathrm{\theta }=3 \\ \therefore 2\mathrm{\theta }={\sec }^{-1} \left(3\right) \\ \Rightarrow \mathrm{\theta }=\frac{1}{2}{\sec }^{-1} \left(3\right) \end{gathered}$$