JEE Main 2024 — 9 April, Shift 1 — Question 15

$\overrightarrow{\mathrm{n}}=\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{q}}$

$\vec{n}=\left| \begin{matrix}\overset{}{\mathop{i}}\, & \overset{}{\mathop{j}}\, & \overset{}{\mathop{k}}\, \\4 & -11 & 5 \\3 & -6 & 1 \\\end{matrix} \right|=19\overset{}{\mathop{i}}\,+11\overset{}{\mathop{j}}\,+9\overset{}{\mathop{k}}\,$

S.d. $=$ projection of $\overrightarrow{\mathrm{AB}}$ on $\vec{n}$

$=\left|\frac{\overrightarrow{A B} \cdot \vec{n}}{|\vec{n}|}\right|=\left|\frac{(2 \hat{i}+16 \hat{j}-3 \hat{k}) \cdot(19 \hat{i}+11 \hat{j}+9 \hat{k})}{\sqrt{361+121+81}}\right|$

$=\frac{38+176-27}{\sqrt{563}}$

S.d. $=\frac{187}{\sqrt{563}}$