JEE Main 2023 (Online) 10th April Evening ShiftQuestion 1
Let $\vec{a}=2\hat{i}+7\hat{j}-\hat{k}$, $\vec{b}=3\hat{i}+5\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}+2\hat{k}$. Let $\vec{d}$ be a vector which is perpendicular to both $\vec{a}$ and $\vec{b}$, and $\vec{c}\cdot\vec{d}=12$. Then $\left(-\hat{i}+\hat{j}-\hat{k}\right)\cdot(\vec{c}\times\vec{d})$ is equal to
Vector Algebra
Solution
$\vec a\times\vec b=(35,-13,-21)$, so $\vec d=k(35,-13,-21)$. $\vec c\cdot\vec d=k(35+13-42)=6k=12\Rightarrow k=2$, $\vec d=(70,-26,-42)$. $\vec c\times\vec d=(94,182,44)$, so $(-1,1,-1)\cdot(94,182,44)=-94+182-44=44$. Option (C).