JEE Main 2021 (Online) 3rd August Evening ShiftQuestion 1
Let $\vec{a}=2\hat{i}+\alpha\hat{j}+\hat{k}$ and $\vec{b}=\beta\hat{i}-5\hat{j}+\gamma\hat{k}$, where $\alpha,\beta$ and $\gamma$ are real numbers. If $\vec{a}\times\vec{b}=26\hat{i}-11\hat{j}-19\hat{k}$, then $\alpha-\beta+\gamma$ is equal to $\_\_\_\_\_\_$ .
Vector Algebra
Answer$7$
Solution
$\vec a\times\vec b=(\alpha\gamma+5,\ \beta-2\gamma,\ -10-\alpha\beta)$. $\alpha\gamma+5=26\Rightarrow\alpha\gamma=21$; $\beta-2\gamma=-11$; $\alpha\beta=9$. $\beta=2\gamma-11$, $\alpha=21/\gamma$: $\tfrac{21}{\gamma}(2\gamma-11)=9\Rightarrow42\gamma-231=9\gamma\Rightarrow\gamma=7$. So $\alpha=3,\ \beta=3$. $\alpha-\beta+\gamma=3-3+7=7$.