If $P(3,2,6)$ is a point in space and $Q$ is a point on the line $\bar{r}=(\hat{i}-\hat{j}+2 \hat{k})+\mu(-3…
If $P(3,2,6)$ is a point in space and $Q$ is a point on the line $\bar{r}=(\hat{i}-\hat{j}+2 \hat{k})+\mu(-3 \hat{i}+\hat{j}+5 \hat{k})$, then the value of $\mu$ for which the vector $\overline{P Q}$ is parallel to the plane $x-4 y+3 z=1$, is
$\frac{1}{4}$
$-\frac{1}{8}$
$\frac{1}{8}$
$-\frac{1}{4}$
Solution
Any point on the vector $\vec{r}$ can be taken as,
$\begin{aligned}
& \mathrm{Q} \equiv\{(1-3 \mu),(\mu-1),(5 \mu+2)\} \text { gives } \\
& \overrightarrow{\mathrm{PQ}}=\{-3 \mu-2, \mu-3,5 \mu-4\}
\end{aligned}$
Now, the $\mathrm{PQ} \overrightarrow{\mathrm{Q}}$ must be perpendicular to the normal for the given plane.
$\begin{aligned}
& 1(-3 \mu-2)-4(\mu-3)+3(5 \mu-4)=0 \\
& \Rightarrow-3 \mu-2-4 \mu+12+15 \mu-12=0 \\
& \Rightarrow 8 \mu=2 \\
& \Rightarrow \mu=\frac{1}{4}
\end{aligned}$