JEE Main 2026 — 28 January, Morning Shift — Question 18

$|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|^{2}+|\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}}|^{2}+|\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}|^{2}=9$

$\Rightarrow \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}}-=-\frac{3}{2}$

$\Rightarrow \vec{a}+\vec{b}+\vec{c}=0 \Rightarrow \vec{b}+\vec{c}=-\vec{a}$

$|2 \overrightarrow{\mathrm{a}}+\mathrm{k}(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}})|=3$

$|\overrightarrow{\mathrm{a}}(2-\mathrm{k})|=3$

$\mathrm{K}=5$ or $-1$

Positive value of k is 5