Let $P(3,2,6)$ be a point in space and $Q$ be a point on the line $\bar{r}=\hat{i}-\hat{j}+2 \hat{k}+\mu(-3…
Let $P(3,2,6)$ be a point in space and $Q$ be 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 $\overline{\mathrm{PQ}}$ is parallel to the plane $x-4 y+3 z=1$ is
$\frac{1}{4}$
$-\frac{1}{4}$
$\frac{1}{8}$
$-\frac{1}{8}$
Solution
Let the position vector of Q be
$\begin{array}{ll}
& (\hat{i}-\hat{j}+2 \hat{k})+\mu(-3 \hat{i}+\hat{j}+5 \hat{k}) \\
& =(-3 \mu+1) \hat{i}+(\mu-1) \hat{j}+(5 \mu+2) \hat{k} \\
\therefore \quad & \overline{P Q}=(-3 \mu-2) \hat{i}+(\mu-3) \hat{j}+(5 \mu-4) \hat{k}
\end{array}$ Since $\overline{\mathrm{PQ}}$ is parallel to the plane,
$\begin{aligned}
& (-3 \mu-2)(1)+(\mu-3)(-4)+(5 \mu-4)(3)=0 \\
& \Rightarrow \mu=\frac{1}{4}
\end{aligned}$