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 $\hat{a}cost+\hat{b}\sin t.$ 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. $\hat{u}=\frac{\hat{a}+\hat{b}}{|\hat{a}+\hat{b}|}andM=(1+\hat{a}.\hat{b}{)}^{1/2}$
• B. $\hat{u}=\frac{\hat{a}-\hat{b}}{|\hat{a}-\hat{b}|}$ 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 $\overrightarrow{OP}at(t=\frac{\pi }{4})=\frac{\hat{a}+\hat{b}}{|\hat{a}+\hat{b}|}$