JEE Main 2024 (Online) 31st January Morning ShiftQuestion 1
Let $\vec{a}=3\hat{i}+\hat{j}-2\hat{k}$, $\vec{b}=4\hat{i}+\hat{j}+7\hat{k}$ and $\vec{c}=\hat{i}-3\hat{j}+4\hat{k}$ be three vectors. If a vector $\vec{p}$ satisfies $\vec{p}\times\vec{b}=\vec{c}\times\vec{b}$ and $\vec{p}\cdot\vec{a}=0$, then $\vec{p}\cdot\left(\hat{i}-\hat{j}-\hat{k}\right)$ is equal to
Vector Algebra
Solution
$\vec p\times\vec b=\vec c\times\vec b\Rightarrow(\vec p-\vec c)\times\vec b=0\Rightarrow\vec p=\vec c+t\vec b=(1+4t,-3+t,4+7t)$. $\vec p\cdot\vec a=3(1+4t)+(-3+t)-2(4+7t)=-8-t=0\Rightarrow t=-8$. $\vec p=(-31,-11,-52)$. $\vec p\cdot(1,-1,-1)=-31+11+52=32$. Option (D).