JEE Main 2025 — 2 April, Evening Shift — Question 30

Eqn. of $L_{1}: 7 \hat{i}+6 \hat{j}+2 \hat{k}+\lambda(-3 \hat{i}+2 \hat{j}+4 \hat{k})$

Eqn. of $L_{2}: 5 \hat{i}+3 \hat{j}+4 \hat{k}+\lambda(2 \hat{i}+\hat{j}+3 \hat{k})$

Shortest distance between $L_{1}$ and $L_{2}$

$=\left|\frac{\left(\vec{a}_{2}-\vec{a}_{1}\right) \cdot\left(\vec{b}_{1} \times \vec{b}_{2}\right)}{\left|\vec{b}_{1} \times \vec{b}_{2}\right|}\right|$

$=\left|\frac{(-2 \hat{i}-3 \hat{j}+2 \hat{k}) \cdot(2 \hat{i}+17 \hat{j}-7 \hat{k})}{\sqrt{2^{2}+(-17)^{2}+(-7)^{2}}}\right|$

$=\left|\frac{-4-51-14}{\sqrt{4+289+49}}\right|$

$=\left|\frac{-69}{\sqrt{342}}\right|=\frac{23}{\sqrt{38}}$