Question

Let $\vec{a}=\alpha \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=2 \hat{i}+\hat{j}-\alpha \hat{k}, \alpha>0$ . If the projection of $\vec{a} \times \vec{b}$ on the vector $-\hat{ i }+2 \hat{ j }-2 \hat{ k }$ is $30$ , then $\alpha$ is equal to

• A. $\frac{15}{2}$
• B. 8
• C. $\frac{13}{2}$
• D. 7

Answer 1

Answer: (D)

$\vec{a} \times \vec{b}=(1-\alpha) \hat{i}+\left(\alpha^2-2\right) \hat{j}+(\alpha-2) \hat{k}$
Projection of $\vec{a} \times \vec{b} \text { on }-\hat{i}+2 \hat{j}-2 \hat{k}$
$=\frac{(\vec{a} \times \vec{b}) \cdot(-\hat{i}+2 \hat{j}-2 \hat{k})}{3}=30$
$\Rightarrow 2 \alpha^2-\alpha-91=0$
$\Rightarrow \alpha=7,-\frac{13}{2}$