Three lines $L_{1}: \vec{r}=\lambda \hat{i}, \lambda \in R, L_{2}: \vec{r}=\hat{k}+\mu \hat{j}, \mu

Question

Three lines $L_{1}: \vec{r}=\lambda \hat{i}, \lambda \in R, L_{2}: \vec{r}=\hat{k}+\mu \hat{j}, \mu \in R$ and $L_{3}: \vec{r}=\hat{i}+\hat{j}+v \hat{k}, v \in R$ are given. For which\ point(s) $Q$ on $L_{2}$ can we find a point $P$ on $L_{1}$ and a point $R$ on $L_{3}$ so that $P, Q$ and $R$ are collinear?\

Professor PC Thomas Classes · Answer

( $\mathrm{a}, \mathrm{d})$ : Let $P(a, 0,0), Q(0, b, 1)$ and $R(1,1, c)$ be points on the line $L_{1}, L_{2}$ and $L_{3}$ respectively.\ $P, Q, R$ are collinear if $\frac{-a}{1}=\frac{b}{1-b}=\frac{1}{c-1}$\ As long as $b 
eq 0,1$, we can have a unique $a$ and $c$. Thus $Q$ can't be collinear at ( $0,1,1$ ) and $(0,0,1)$.\