Find the value of $\displaystyle p$ for which the lines \[\begin{aligned} & \vec{r}=\lambda \hat{i}+(2 \lambda+1) \hat{j}+(3 \lambda+2) \hat{k} \text { and } \\ & \vec{r}=\hat{i}-3 \mu \hat{j}+(p \mu+7) \hat{k} \end{aligned} \] are perpendicular to each other and also intersect. Also, find the point of intersection of the given lines.

Given lines are $$ \begin{aligned} & \vec{r}=\hat{\jmath}+2 \hat{k}+\lambda(\hat{\imath}+2 \hat{\jmath}+3 \hat{k}) \text { and } & \vec{r}=\hat{\imath}+7 \hat{k}+\mu(-3 \hat{\jmath}+p \hat{k}) \end{aligned}The lines are perpendicular(\hat{\imath}+2 \hat{\jmath}+3 \hat{k}) \cdot(-3 \hat{\jmath}+p \hat{k})=0 \Rightarrow-6+3 p=0 \Rightarrow p=2The coordinates of any point on the two lines are(\lambda, 2 \lambda+1,3 \lambda+2) \text { and }(1,-3 \mu, 2 \mu+7)For the point of intersection, we must have $\displaystyle \lambda=1,2 \lambda+1=-3 \mu, 3 \lambda+2=2 \mu+7$ for some $\displaystyle \lambda$ and $\displaystyle \mu$. Solving first two equations, we get, $\displaystyle \lambda=1, \mu=-1$, which satisfies the third equation as $\displaystyle 5=5$ is true. Hence, the lines intersect each other. The Point of intersection is ( $\displaystyle 1,3,5$ )