A line \(\mathrm{L}_1\) passes through the point, whose p. v. (position vector) \(3 \hat{\mathrm{i}}\), is parallel to the vector \(-\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\). Another line \(\mathrm{L}_2\) passes through the point having p.v. \(\hat{i}+\hat{j}\) is parallel to vector \(\hat{i}+\hat{k}\), then the point of intersection of lines \(L_1\) and \(L_2\) has p.v.

Equation of line \(\mathrm{L}_1\) is \(\overline{\mathrm{r}}=3 \hat{\mathrm{i}}+\lambda(-\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})\)
Equation of line \(L_2\) is \(\overline{r^{\prime}}=\hat{i}+\hat{j}+\lambda^{\prime}(\hat{i}+\hat{k})\)
The point of intersection of \(\mathrm{L}_1\) and \(\mathrm{L}_2\) will satisfy \(\overline{\mathrm{r}}=\overline{\mathrm{r}^{\prime}}\)
\(
\begin{aligned}
& \Rightarrow 3 \hat{\mathrm{i}}+\lambda(-\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\lambda^{\prime}(\hat{\mathrm{i}}+\hat{\mathrm{k}}) \\
& \Rightarrow(3-\lambda) \hat{\mathrm{i}}+\lambda \hat{\mathrm{j}}+\lambda \hat{\mathrm{k}}=\left(1+\lambda^{\prime}\right) \hat{\mathrm{i}}+\hat{\mathrm{j}}+\lambda^{\prime} \hat{\mathrm{k}} \\
& \Rightarrow 3-\lambda=1+\lambda^{\prime} \text { and } \lambda=1 \\
& \Rightarrow \lambda=1 \text { and } \lambda^{\prime}=1
\end{aligned}
\)
Substituting the value of \(\lambda\) in (i), we get the point of intersection.
\(\therefore\) The point of intersection of lines \(\mathrm{L}_1\) and \(\mathrm{L}_2\) has p.v. \(2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}\).