JEE Main 2021 (Online) 4th August Evening ShiftQuestion 1
Let $\vec{a}=-\hat{i}+\hat{j}$ and $\vec{b}=\hat{i}+\hat{j}-\hat{k}$ be two given vectors. If a vector $\vec{c}$ satisfies $(\vec{a}\times\vec{c})+\vec{b}=\vec{0}$ and $\vec{a}\cdot\vec{c}=3$, then $10|\vec{c}|^{2}$ is equal to $\_\_\_\_\_\_\_\_$ .
Vector Algebra
Answer$60$
Solution
$\vec a\times(\vec a\times\vec c)=(\vec a\cdot\vec c)\vec a-|\vec a|^2\vec c=-\vec a\times\vec b$, so $3\vec a-2\vec c=(1,1,2)$ and $\vec c=(-2,1,-1)$ (check $\vec a\cdot\vec c=3$). $10|\vec c|^2=60$.