MCQ
The distance between the lines
$\overline{ r }=-\hat{ i }+3 \hat{ j }+\hat{ k }+\lambda(5 \hat{ i }+\hat{ j }+4 \hat{ k })$ and
$\overline{r}=3 \hat{i}+\hat{j}+\mu(5 \hat{i}+\hat{j}+4 \hat{k})$ is
  • $\frac{7}{\sqrt{3}}$
  • B
    $\frac{14}{\sqrt{3}}$
  • C
    $\sqrt{3}$
  • D
    $\frac{7}{\sqrt{6}}$

Answer

Correct option: A.
$\frac{7}{\sqrt{3}}$
(A)
Comparing the given equations with
$\overline{ r }=\overline{ a }_1+\lambda \overline{ b }_1$ and
$\overline{ r }=\overline{ a }_2+\lambda \overline{ b }_2$ we get
$\overline{a_1}--\hat{i}+3 \hat{j}+\hat{k}$, and $\bar{a}=-3 \hat{i}+\hat{j}$
$\overline{ b }_1=\overline{ b }_2=\overline{ b }=5 \hat{ i }+\hat{ j }+4 \hat{ k }$
$\therefore \quad$ The lines are parallel
$\bar{a}_2-\bar{a}_1=4 \hat{i}-2 \hat{j}-\hat{k}$
$\left(\bar{a}_2-\bar{a}_1\right) \times \bar{b}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 4 & -2 & -1 \\ 5 & 1 & 4\end{array}\right|$
$=\hat{ i }(-8+1)-\hat{ j }(16+5)+\hat{ k }(4+10)$
$=-7 \hat{ i }-21 \hat{ j }+14 \hat{ k }$
$\therefore \quad$ The distance between the parallel lines is
$d =\left|\frac{\left(\overline{ a }_2-\overline{ a }_1\right) \times \overline{ b }}{|\overline{ b }|}\right|$
$\therefore \quad d=\left|\frac{-7 \hat{i}-21 \hat{j}+14 \hat{k}}{\sqrt{25+1+16}}\right|$
$=\sqrt{\frac{49+441+196}{42}}$
$=\frac{7}{\sqrt{3}}$

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