JEE Main 2026 — 24 January, Morning Shift — Question 5

$\quad \overrightarrow{\mathrm{c}}=\left|\begin{array}{ccc}\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\2 & 1 & -2\\ 1 & 1 & 0\end{array}\right|$

$\overrightarrow{\mathrm{c}}=2 \hat{\mathrm{i}}+2 \hat{\mathrm{k}}+\hat{\mathrm{k}},|\mathrm{c}|=3$

$|\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{d}}|=3$

$|\overrightarrow{\mathrm{c}}||\overrightarrow{\mathrm{d}}| \sin \frac{\pi}{4}=3 \Rightarrow|\overrightarrow{\mathrm{~d}}|=\sqrt{2}$

$|\overrightarrow{\mathrm{d}}-\overrightarrow{\mathrm{a}}|=\sqrt{11}$

$\Rightarrow|\overrightarrow{\mathrm{a}}|^{2}+|\overrightarrow{\mathrm{d}}|^{2}-2 \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{d}}=11$ $9+2-2 \vec{a} \cdot \vec{d}=11$

$\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{d}}=0$