JEE Main 2024 — 6 April, Shift 1 — Question 30

$\vec{a} \times \vec{b}+\vec{a} \times \vec{c}+\vec{b} \times \vec{c}=(1,8,13)$

$\vec{a} \times(\vec{a} \times \vec{b})+\vec{a} \times(\vec{a} \times \vec{c})+\vec{a} \times(\vec{b} \times \vec{c})$

$=\overrightarrow{\mathrm{a}} \times(\hat{\mathrm{i}}+8 \hat{\mathrm{j}}+13 \hat{\mathrm{k}})$

$(\vec{a} \cdot \vec{b}) \vec{a}-a^{2} \vec{b}+(\vec{a} \cdot \vec{c}) \vec{a}-a^{2} \vec{c}+(\vec{a} \cdot \vec{c}) \vec{b}-(\vec{a} \cdot \vec{b}) \vec{c}=\vec{a} \times(\hat{i}+8 \hat{j}+13 \hat{k})$

$\Rightarrow-26 \vec{a}-29 \vec{b}+13 \vec{a}-29 \overrightarrow{\mathrm{c}}+13 \vec{b}+26 \vec{c}=\vec{a} \times(\hat{i}+8 \hat{j}+13 \hat{k})$

$$\begin{aligned} & \Rightarrow -13 \vec{a} - 16 \vec{b} + 3 \vec{c} = \vec{a} \times (\vec{i} + 8\vec{j} + 13\vec{k}) \\ & \Rightarrow -13 \vec{a} \cdot \vec{b} - 16 \, b^2 + 3 \vec{b} \cdot \vec{c} = \big(\vec{a} \times (\vec{i} + 8\vec{j} + 13\vec{k})\big) \cdot \vec{b} \\ & \Rightarrow (-13)(-26) - 16(50) - 3 \vec{b} \cdot \vec{c} = \begin{vmatrix} 2 & -3 & 4 \\ 1 & 8 & 13 \\ 3 & 4 & -5 \end{vmatrix} \\ & \Rightarrow -462 - 3 \vec{b} \cdot \vec{c} = -396 \\ & \Rightarrow \vec{b} \cdot \vec{c} = -22 \end{aligned}$$ $$\therefore 24 - \vec{b} \cdot \vec{c} = 24 - (-22) = 46$$