JEE Main 2021 (Online) 25th February Morning ShiftQuestion 1
Let $\vec{a}=\hat{i}+2\hat{j}-\hat{k}$, $\vec{b}=\hat{i}-\hat{j}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $\vec{r}\times\vec{a}=\vec{c}\times\vec{a}$ and $\vec{r}\cdot\vec{b}=0$, then $\vec{r}\cdot\vec{a}$ is equal to $\_\_\_\_\_\_$ .
Vector Algebra
Answer$12$
Solution
$(\vec r-\vec c)\times\vec a=\vec0\Rightarrow\vec r=\vec c+\lambda\vec a=(1+\lambda,\,-1+2\lambda,\,-1-\lambda)$. $\vec r\cdot\vec b=(1+\lambda)-(-1+2\lambda)=2-\lambda=0\Rightarrow\lambda=2$, so $\vec r=(3,3,-3)$. $\vec r\cdot\vec a=3+6+3=12$.