JEE Main 2025 — 22 January, Morning Shift — Question 25

$\vec{c}=\left(\frac{\vec{b} \cdot \vec{a}}{|\vec{a}|}\right) \frac{\vec{a}}{| | \vec{a} \mid}$

$=\left(\frac{\lambda+8}{9}\right)(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})$

$|\vec{a}+\vec{c}|=7 \Rightarrow \lambda=4$

Area of parallelogram

$\left.=|\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}|=\left|\begin{array}{ccc}\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{k}\\ \frac{4}{3} & \frac{8}{3} & \frac{8}{3}\\ 4 & 0 & 4\end{array}\right| \right\rvert\,$

$=16$