Determine if the lines r =( i + j - k )+λ(3 i - j ) and r =(4 i - k )+μ(2 i +3 k ) intersect with each other.

\[\begin{aligned} & \vec{b}_{1}=3 \hat{\mathrm{i}}-\hat{\mathrm{j}}, \vec{b}_{2}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{k}}, \overrightarrow{\mathrm{a}_{2}}=4 \hat{\mathrm{i}}-\hat{\mathrm{k}}, \vec{\mathrm{a}}_{1}=\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}} \\ & \left(\overrightarrow{\mathrm{a}_{2}}-\vec{\mathrm{a}}_{1}\right)=3 \hat{\mathrm{i}}-\hat{\mathrm{j}} \\ & \vec{b}_{1} \times \vec{b}_{2}=-3 \hat{\mathrm{i}}-9 \hat{\mathrm{j}}+2 \hat{\mathrm{k}} \end{aligned} \]