Question

Let $\overset{ \rightarrow }{V}\left(\theta \right)=\left(cos \theta + sec ⁡ \theta \right)\hat{a}+\left(cos ⁡ \theta - s e c \theta \right)\hat{b}$ , where $\hat{a}$ and $\hat{b}$ are unit vectors and the angle between $\hat{a}$ and $\hat{b}$ is $60^{o}$ , then the minimum value of $\left|\overset{ \rightarrow }{V}\right|^{4}$ is equal to

Answer 1

$\left(\left|\overset{ \rightarrow }{V}\right|\right)^{2}=\left(\left|\left(cos \theta + sec ⁡ \theta \right) \hat{a} + \left(cos ⁡ \theta - sec ⁡ \theta \right) \hat{b}\right|\right)^{2}$
$=\left(cos \theta + sec ⁡ \theta \right)^{2}+\left(cos ⁡ \theta - sec ⁡ \theta \right)^{2}+\left(\right.cos ⁡ \theta - sec ⁡ \theta \left.\right)\left(\right.cos ⁡ \theta + sec ⁡ \theta \left.\right)$
$=3cos^{2} \theta +sec^{2} ⁡ \theta \geq 2 \sqrt{3}$
$\Rightarrow $ least value of $\left|\overset{ \rightarrow }{V}\right|^{2}=2\sqrt{3}$
$\Rightarrow $ least value of $\left|\overset{ \rightarrow }{V}\right|^{4}=12$