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 $L_3: \bar{r}=\hat{i}+\hat{j}+v \hat{k}, v \in R$ 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?

• A. $\hat{k}+\frac{1}{2} \hat{j}$
• B. $\hat{k}+\hat{j}$
• C. $\hat{k}$
• D. None of these

Answer 1

Answer: (A)

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 $b \neq 0,1$,
we can have a unique $a$ and $c$.
Thus $Q$ can't be collinear at $(0,1,1)$ and $(0,0,1)$.