Let a=i+2j+k and b=2i+7j+3k.

Question

Let $\vec{a}=\hat{i}+2\hat{j}+\hat{k}$ and $\vec{b}=2\hat{i}+7\hat{j}+3\hat{k}$. Let ${L}_{1}:\overset{	o }{r}=(-\hat{i}+2\hat{j}+\hat{k})+\lambda \vec{a},\lambda \in R$ and ${L}_{2}:\overset{	o }{r}=(\hat{j}+\hat{k})+\mu \vec{b},\mu \in R$ be two lines. If the line ${L}_{3}$ passes through the point of intersection of ${L}_{1}$ and ${L}_{y}$ and is parallel to $\vec{a}+\vec{b}$, then ${L}_{3}$ passes through the point :

Accepted Answer

$\begin{gathered} {L}_{1}:\overset{	o }{r}=(-\hat{i}+2\hat{j}+\hat{k})+\lambda (\hat{i}+2\hat{j}+\hat{k}) \ \Rightarrow \overset{	o }{r}=(\lambda -1)\hat{i}+2(\lambda +1)\hat{j}+(\lambda +1)\hat{k} \ {L}_{2}:\overset{	o }{r}=(\hat{j}+\hat{k})+\mu (2\hat{i}+7\hat{j}+3\hat{k}) \ \Rightarrow \overset{	o }{r}=2\mu \hat{i}+(1+7\mu )\hat{j}+(1+3\mu )\hat{k} \end{gathered}$

For point of intersection equating respective components

$\begin{gathered} \Rightarrow \lambda -1=2\mu \, \ 2(\lambda +1)=1+7\mu \, \ \lambda +1=1+3\mu \, \ \end{gathered}$

$\begin{gathered} \Rightarrow \lambda =3\mu =1 \ \Rightarrow \overset{	o }{a}+\overset{	o }{b}=3\hat{i}+9\hat{j}+4\hat{k} \ {L}_{3}:\overset{	o }{r}=2\hat{i}+8\hat{j}+4\hat{k}+\alpha (3\hat{i}+9\hat{j}+4\hat{k}) \ \alpha =2,\overset{	o }{r}=8\hat{i}+26\hat{j}+12\hat{k} \end{gathered}$

Solutions · 1