Line and Plane — Maths STD 12 Science — Question

(c) : Let $\vec{a}$ and $\vec{b}$ be the position vectors of the points $P(6,-1,2)$ and $R(5,2,4)$.
Thus, $\vec{a}=6 \hat{i}-\hat{j}+2 \hat{k}, \vec{b}=5 \hat{i}+2 \hat{j}+4 \hat{k}$
Now, $\vec{b}-\vec{a}=-\hat{i}+3 \hat{j}+2 \hat{k}$
$\therefore \quad$ Vector equation of the line passing through $P$ and $R$
$\begin{aligned}
& \text { is } \vec{r}=\vec{a}+\mu \vec{b} \Rightarrow \vec{r}=(6 \hat{i}-\hat{j}+2 \hat{k})+\mu(-\hat{i}+3 \hat{j}+2 \hat{k}) \\
& \Rightarrow \vec{r}=(6-\mu) \hat{i}+(3 \mu-1) \hat{j}+(2 \mu+2) \hat{k}
\end{aligned}
$
This equation passes through $Q(8,-7,2 \lambda)$.
$\begin{aligned}
& \therefore 8 \hat{i}-7 \hat{j}+2 \lambda \hat{k}=(6-\mu) \hat{i}+(3 \mu-1) \hat{j}+(2 \mu+2) \hat{k} \\
& \Rightarrow 6-\mu=8,3 \mu-1=-7,2 \mu+2=2 \lambda \\
& \Rightarrow \mu=-2 \Rightarrow 2(-2)+2=2 \lambda \Rightarrow \lambda=-1
\end{aligned}$