JEE Main 2026 — 23 January, Evening Shift — Question 3

$\vec{a}=\hat{i}-2 \hat{j}+3 \hat{k}, \vec{b}=2 \hat{i}+\hat{j}-\hat{k}, \vec{c}=\lambda \hat{i}+\hat{j}+\hat{k}$, and $\vec{v}=\vec{a} \times \vec{b}$.

If $\vec{v} \cdot \vec{c}=11$

$\vec{v}=(\vec{a} \times \vec{b})=(-\hat{i}+7 \hat{j}+5 \hat{k})$

$\vec{v} \cdot \vec{c}=11=(-\hat{i}+7 \hat{j}+5 \hat{k}) \cdot(\lambda \hat{i}+\hat{j}+\hat{k})=11$

$\Rightarrow-\lambda+7+5=11$

$\Rightarrow \lambda=1$

Length of projection of $\vec{b}$ on $\vec{c}=\vec{b} \cdot \hat{c}$

$\Rightarrow p=\left|(2 \hat{i}+\hat{j}-\hat{k}) \cdot \frac{(\hat{i}+\hat{j}+\hat{k})}{\sqrt{3}}\right|=\frac{2+1-1}{\sqrt{3}}=\frac{2}{\sqrt{3}}$

$\Rightarrow 9 \mathrm{p}^{2}=9\left(\frac{4}{3}\right)=12$