Mathematics - Vector Question with Solution | TestHub

As given
$L_1\Rightarrow \overrightarrow{r}=\lambda i, \lambda \in R$
$L_2\Rightarrow \overrightarrow{r}=k+\mu j, \mu \in R$
and$L_3\Rightarrow \overrightarrow{r}=i+j+\nu k, \nu \in R$
Let point$P\left(\lambda , 0, 0\right), Q\left(0, \mu , 1\right)$and$R\left(1, 1, \nu \right)$
and$\overrightarrow{PQ}=-\lambda \hat{i}+\mu \hat{j}+\hat{k}, \overrightarrow{PR}=\left(1-\lambda \right)\hat{i}+\hat{j}+\nu \hat{k}$
as$P, Q$and$R$are co-linear
$\Rightarrow \overrightarrow{PQ}$is parallel to$\overrightarrow{PR}$
$\Rightarrow -\frac{\lambda }{1-\lambda }=\frac{\mu }{1}=\frac{1}{\nu }$
$\Rightarrow \lambda =\frac{\mu }{\mu -1}$and$\nu =\frac{1}{\mu }$
hence$\begin{array}{cc}\mu \ne 0& \mu \ne 1\end{array}$
$Q\ne \left(0, 0, 1\right)$and$Q\ne \left(0, 1, 1\right)$
$Q\ne \hat{k} \& Q\ne \left(\hat{j}+\hat{k}\right)$