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