Find the value of p for which the lines r =λ i +(2 λ+1) j +(3 λ+2) k and r = i -3 μ j +(p μ+7) k are perpendicular to each other and also intersect.…

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 $$(\hat{\imath}+2 \hat{\jmath}+3 \hat{k}) \cdot(-3 \hat{\jmath}+p \hat{k})=0 \Rightarrow-6+3 p=0 \Rightarrow p=2 $$ The 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 ( $\displaystyle 1,3,5$ )