Question

Let $\overrightarrow{ a }=\alpha \hat{ i }+3 \hat{ j }-\hat{ k }, \overrightarrow{ b }=3 \hat{ i }-\beta \hat{ j }+4 \hat{ k } $ and $\overrightarrow{ c }=\hat{ i }+2 \hat{ j }-2 \hat{ k }$ where $\alpha, \beta \in R$, be three vectors. If the projection of $\overrightarrow{ a }$ on $\overrightarrow{ c }$ is $\frac{10}{3}$ and $\overrightarrow{ b } \times \overrightarrow{ c }=-6 \hat{ i }+10 \hat{ j }+7 \hat{ k }$, then the value of $\alpha+\beta$ equal to :

• A. 3
• B. 4
• C. 5
• D. 6

Answer 1

Answer: (A)

$\frac{\overrightarrow{ a } \cdot \overrightarrow{ c }}{|\overrightarrow{ c }|}=\frac{10}{3}$
$\Rightarrow \frac{\alpha+6+2}{\sqrt{1+4+4}}=\frac{10}{3} \Rightarrow \alpha=2$
and $\left|\begin{array}{ccc}\hat{i} & \hat{ j } & \hat{ k } \\ 3 & -\beta & 4 \\ 1 & 2 & -2\end{array}\right|=-6 \hat{ i }+\hat{ j }+\hat{ k }$
$\Rightarrow 2 \beta-8=-6 \Rightarrow \beta=1$
$\Rightarrow \alpha+\beta=3$