JEE Main 2024 (Online) 5th April Evening ShiftQuestion 1
Let $\vec{a}=2\hat{i}+5\hat{j}-\hat{k}$, $\vec{b}=2\hat{i}-2\hat{j}+2\hat{k}$ and $\vec{c}$ be three vectors such that $\left(\vec{c}+\hat{i}\right)\times\left(\vec{a}+\vec{b}+\hat{i}\right)=\vec{a}\times\left(\vec{c}+\hat{i}\right)$. If $\vec{a}\cdot\vec{c}=-29$, then $\vec{c}\cdot\left(-2\hat{i}+\hat{j}+\hat{k}\right)$ is equal to :
Vector Algebra
Solution
$(\vec c+\hat i)\times(\vec a+\vec b+\hat i)+(\vec c+\hat i)\times\vec a=\vec0$, so $\vec c+\hat i=\lambda(2\vec a+\vec b+\hat i)=\lambda(7,8,0)$. $\vec a\cdot\vec c=54\lambda-2=-29$ gives $\lambda=-\frac12$, $\vec c=\left(-\frac92,-4,0\right)$ and $\vec c\cdot(-2,1,1)=9-4=5$.