Question

If unit vector $\vec{c}$ makes an angle $\frac{\pi}{3}$ with $\hat{i}+\hat{j}$, then minimum and maximum values of $\left(\hat{i}\times\hat{j}\right)\cdot\vec{c}$ respectively are

• A. $0, \frac{\sqrt{3}}{2}$
• B. $-\frac{\sqrt{3}}{2}, \frac{\sqrt{3}}{2}$
• C. $-1, \frac{\sqrt{3}}{2}$
• D. None of these

Answer 1

Answer: (B)

Angle between $\hat{i}+\hat{j}$ and $\left(\hat{i}\times\hat{j}\right)$ is $\frac{\pi}{2}$
$\Rightarrow $ Angle between $\vec{c}$ and $\left(\hat{i}\times\hat{j}\right) \ge \frac{\pi}{2}-\frac{\pi }{3} = \frac{\pi }{6}$
$\Rightarrow \left(\hat{i}\times \hat{j}\right)\cdot\vec{c} \le \left|\hat{i}\times \hat{j}\right|\left|\vec{c}\right|cos \frac{\pi }{6}$
$\Rightarrow -\frac{\sqrt{3}}{2} \le \left(\hat{i}\times \hat{j}\right)\cdot \vec{c} \le \frac{\sqrt{3}}{2}$