JEE Main 2021 (Online) 27th July Evening ShiftQuestion 1
Let $\vec a=\hat i-\alpha\hat j+\beta\hat k$, $\vec b=3\hat i+\beta\hat j-\alpha\hat k$ and $\vec c=-\alpha\hat i-2\hat j+\hat k$, where $\alpha$ and $\beta$ are integers. If $\vec a\cdot\vec b=-1$ and $\vec b\cdot\vec c=10$, then $(\vec a\times\vec b)\cdot\vec c$ is equal to ________.
Vector Algebra
Answer$9$
Solution
$\vec a\cdot\vec b=3-2\alpha\beta=-1\Rightarrow\alpha\beta=2$. $\vec b\cdot\vec c=-3\alpha-2\beta-\alpha=10\Rightarrow2\alpha+\beta=-5$. Then $2\alpha^2+5\alpha+2=0\Rightarrow\alpha=-2$ (integer), $\beta=-1$. $\vec a=(1,2,-1),\ \vec b=(3,-1,2),\ \vec c=(2,-2,1)$; $\vec a\times\vec b=(3,-5,-7)$. $(\vec a\times\vec b)\cdot\vec c=6+10-7=9$.