JEE Main 2025 — 23 January, Evening Shift — Question 18

$$\vec{a} = (2, 1, -3)$$ $$\vec{b} = (-1, -3, -5)$$ $$\vec{p} \times \vec{q} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & -3 \\ 2 & 4 & -5 \end{vmatrix}$$

$=2 \hat{i}-\hat{j}$

$\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}=-3 \hat{\mathrm{i}}-4 \hat{\mathrm{j}}-2 \hat{\mathrm{k}}$

$\mathrm{S}_{\mathrm{d}}=\frac{|(\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{a}}) \cdot(\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{q}})|}{|\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{q}}|}$.

$=\frac{2}{\sqrt{5}}$ $\left(\mathrm{S}_{\mathrm{d}}\right)^{2}=\frac{4}{5}$

$\mathrm{m}=4, \mathrm{n}=5 \Rightarrow \mathrm{~m}+\mathrm{n}=9$