JEE Main 2026 — 28 January, Morning Shift — Question 24

$\overrightarrow{\mathrm{PQ}}=-2 \hat{i}-\hat{j}+2 \hat{k}$

$\overrightarrow{\mathrm{PR}}=a \hat{\mathrm{i}}+b \hat{\mathrm{j}}-4 \hat{\mathrm{k}} \quad(a, b \in Z)$

$\overrightarrow{\mathrm{PS}}=\hat{\mathrm{i}}-7 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$

$|\overrightarrow{\mathrm{PR}}|=9$ $\mathrm{a}^{2}+\mathrm{b}^{2}+16=81$

$\begin{gathered} \mathrm{a}^{2}+\mathrm{b}^{2}=65 \end{gathered}$ $\cos \theta=\frac{\overrightarrow{\mathrm{PQ}} \cdot \overrightarrow{\mathrm{PS}}}{|\overrightarrow{\mathrm{PQ}}||\overrightarrow{\mathrm{PS}}|}$

$=\frac{-2+7+4}{3.3 \sqrt{6}}=\frac{9}{3.3 \sqrt{6}}$ $=\frac{1}{\sqrt{6}}$

$\frac{1}{\sqrt{6}}=\frac{\overrightarrow{\mathrm{PS}} \cdot \overrightarrow{\mathrm{PR}}}{|\overrightarrow{\mathrm{PS}}||\overrightarrow{\mathrm{PR}}|}=\frac{a-7 b-8}{3 \sqrt{6} \cdot 9}$

$$\begin{gathered} \mathrm{a}-7 \mathrm{~b}=35 \end{gathered}$$

From & (2) $\Rightarrow \mathrm{a}=7, \mathrm{~b}=-4$

$\therefore 3 a-4 b=21+16=37$