JEE Main 2026 — 5 April, Morning Shift — Question 37

$\vec{r} \times \vec{a}-\vec{b} \times \vec{a}=\overrightarrow{0}$ $(\vec{r}-\vec{b}) \times \vec{a}=\overrightarrow{0}$ $\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{b}}=\lambda \overrightarrow{\mathrm{a}}$ $\overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{b}}+\lambda \overrightarrow{\mathrm{a}}$ $\vec{r} \cdot \vec{a}=0 \Rightarrow \vec{a} \cdot \vec{b}+\lambda|\vec{a}|^{2}=0$ $\lambda=-\frac{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}}{|\overrightarrow{\mathrm{a}}|^{2}}=-\frac{(1-2)}{9}=\frac{1}{9}$ $\overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{b}}+\frac{\overrightarrow{\mathrm{a}}}{9}$ $|3 \vec{r}|^{2}=9|\vec{r}|^{2}=9\left(b^{2}+\frac{a^{2}}{81}+\frac{2(\vec{a} \cdot \vec{b})}{9}\right)=44$