Question

Let two non-collinear unit vectors $\hat{a}$ and $\hat{b}$ form an acute angle. A point $P$ moves, so that at any time $t$ the position vector $\overrightarrow{OP}$ (where, $O$ is the origin) is given by $\hat {a}\, cos\, t +\hat{b}\, \sin t.$ When $P$ is farthest from origin $O$ , let $M$ be the length of $\overrightarrow{OP} $ and $\hat{u}$ be the unit vector along $\overrightarrow{OP} .$ Then,

• A. $\hat{u}=\frac{\hat{a}+\hat{b}}{|\hat{a}+\hat{b}|} \,and\, M =(1+\hat{a}.\hat{b})^{1/2}$
• B. $\hat{u}=\frac{\hat{a}-\widehat{b}}{|\hat{a}-\hat{b}|} $ and $M \,=(1+\hat{a}.\hat{b})^{1/2}$
• C. $\hat{u}=\frac{\hat{a}+\hat{b}}{|\hat{a}+\hat{b}|} \, $ and $\, M =(1+2\hat{a}.\hat{b})^{1/2}$
• D. $\hat{u}=\frac{\hat{a}-\hat{b}}{|\hat{a}-\hat{b}|} $ and $M \,=(1+2\hat{a}.\hat{b})^{1/2}$

Answer 1

Answer: (A)

Given, $\overrightarrow{OP}=\hat{a} \cos\, t+\hat{b} \sin t$
$\Rightarrow | \overrightarrow{OP} |=\sqrt{(\hat{a}.\hat{a}) \cos^2 t+(\hat{b}.\hat{b}) \sin^2 t+2\hat{a}.\hat{b} \sin t \cos t}$
$\Rightarrow | \overrightarrow{OP} |=\sqrt{1+\hat{a}.\hat{b} \sin 2t}$
$\Rightarrow | \overrightarrow{OP} |_{max}=M=\sqrt{1+\hat{a}.\hat{b}}$ at $\sin 2t =1\Rightarrow t=\frac{\pi}{4} $
At $ t=\frac{\pi}{4}, \overrightarrow{OP}=\frac{1}{\sqrt{2}}(\hat{a}+\hat{b})$
Unit vector along $\overrightarrow{OP}at \Big(t=\frac{\pi}{4}\Big)=\frac{\hat{a}+\hat{b}}{| \hat{a}+\hat{b} |}$