Let \(P(3,2,6)\) be a point in space and \(Q\) be a point on the line \(\mathbf{r}=(\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}})+\mu(-3 \hat{\mathbf{i}}+\hat{\mathbf{j}}+5 \hat{\mathbf{k}})\).
Then, the value of \(\mu\) for which the vector \(\mathbf{P Q}\) is parallel to the plane \(x-4 y+3 z=1\) is

\(\mathbf{O Q}=(1-3 \mu) \mathbf{i}+(\mu-1) \mathbf{j}+(5 \mu+2) \mathbf{k}\) and \(\mathbf{O P}=3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}\), where \(O\) is origin.
\[
\begin{aligned}
& \text { Now, } \mathbf{P Q}=(1-3 \mu-3) \hat{\mathbf{i}}+(\mu-1-2) \hat{\mathbf{j}} \\
& +(5 \mu+2-6) \hat{\mathbf{k}} \\
& =(2-3 \mu) \hat{\mathbf{i}}+(\mu-3) \hat{\mathbf{j}}+(5 \mu-4) \hat{\mathbf{k}} \\
&
\end{aligned}
\]
\(\because \mathbf{P Q}\) is parallel to the plane
\[
\begin{array}{rr}
& x-4 y+3 z=1 . \\
\therefore & -2-3 \mu-4 \mu+12+15 \mu-12=0 \\
\Rightarrow & 8 \mu=2 \Rightarrow \mu=\frac{1}{4}
\end{array}
\]