JEE Main 2024 — 1 February, Shift 1 — Question 8

$\vec{a}=-5 \hat{i}+j-3 \hat{k}$

$\vec{b}=\hat{i}+2 \hat{j}-4 \hat{k}$

$(\vec{a} \times \vec{b}) \times \hat{i}=(\vec{a} \cdot \hat{i}) \vec{b}-(\vec{b} \cdot \hat{i}) \vec{a}$

$=-5 \vec{b}-\vec{a}$

$=(((-5 \vec{b}-\vec{a}) \times \hat{i}) \times \hat{i})$

$=((-11 \hat{j}+23 \hat{k}) \times \hat{i}) \times \hat{i}$

$\Rightarrow(11 \hat{k}+23 \hat{j}) \times \hat{i}$

$\Rightarrow(11 \hat{j}-23 \hat{k})$

$\vec{C} \cdot(-\hat{i}+\hat{j}+\hat{k})=11-23=-12$