MCQ
The line $L_{1}$ is parallel to the vector $\overrightarrow{\mathrm{a}}=-3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$ and passes through the point $(7,6,2)$ and the line $L_{2}$ is parallel to the vector $\overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ and passes through the point $(5,3,4)$. The shortest distance between the lines $L_{1}$ and $L_{2}$ is :
  • A
    $\frac{23}{\sqrt{38}}$
  • B
    $\frac{21}{\sqrt{57}}$
  • C
    $\frac{23}{\sqrt{57}}$
  • D
    $\frac{21}{\sqrt{38}}$

Answer

A. $\frac{23}{\sqrt{38}}$
$L_{1}:(7 \hat{\mathrm{i}}+6 \hat{\mathrm{j}}+2 \mathrm{k})+\lambda(-3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+4 \mathrm{k})$
$L_{2}:(5 \hat{i}+3 \hat{j}+4 \mathrm{k})+\lambda(2 \hat{i}+\hat{j}+3 \mathrm{k})$
Distance between skew lines
$=\frac{(2 \hat{i}+3 \hat{j}-2 \hat{k}) \cdot(2 \hat{i}+17 \hat{j}-7 \hat{k})}{\sqrt{342}}$
$=\frac{69}{\sqrt{342}}=\frac{69}{3 \sqrt{38}}=\frac{23}{\sqrt{38}}$

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