JEE Main 2022 (Online) 29th June Morning ShiftQuestion 1
Let $\vec{a}=\alpha\hat{i}+3\hat{j}-\hat{k}$, $\vec{b}=3\hat{i}-\beta\hat{j}+4\hat{k}$ and $\vec{c}=\hat{i}+2\hat{j}-2\hat{k}$ where $\alpha,\beta\in\mathbf{R}$, be three vectors. If the projection of $\vec{a}$ on $\vec{c}$ is $\frac{10}{3}$ and $\vec{b}\times\vec{c}=-6\hat{i}+10\hat{j}+7\hat{k}$, then the value of $\alpha+\beta$ is equal to :
Vector Algebra
Solution
Projection of $\vec a$ on $\vec c$ ($|\vec c|=3$): $\frac{\alpha+6+2}{3}=\frac{10}{3}\Rightarrow\alpha=2$. $\vec b\times\vec c$ has $\hat i$-component $(-\beta)(-2)-4\cdot2=2\beta-8=-6\Rightarrow\beta=1$. The other components give $4\cdot1-3\cdot(-2)=10$ and $6+\beta=7$, both consistent. $\alpha+\beta=3$. Answer: (A).