Question

Let two non-collinear unit vectors $\hat{a}\text{and}\hat{b}$ 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. $\hat{u}=\frac{\hat{a}+\hat{b}}{|\hat{a}+\hat{b}|}\text{and}M=(1+\hat{a}.\hat{b}{)}^{\frac{1}{2}}$
  
• B. $\hat{u}=\frac{\hat{a}-\hat{b}}{|\hat{a}-\hat{b}|}\text{and}M=(1+\hat{a}.\hat{b}{)}^{\frac{1}{2}}$
• C. $\hat{u}=\frac{\hat{a}+\hat{b}}{|\hat{a}+\hat{b}|}\text{and}M=(1+2\hat{a}.\hat{b}{)}^{\frac{1}{2}}$
• D. $\hat{u}=\frac{\hat{a}-\hat{b}}{|\hat{a}-\hat{b}|}\text{and}M=(1+2\hat{a}.\hat{b}{)}^{\frac{1}{2}}$

Answer 1

The correct option is A

$\hat{u}=\frac{\hat{a}+\hat{b}}{|\hat{a}+\hat{b}|}\text{and}M=(1+\hat{a}.\hat{b}{)}^{\frac{1}{2}}$

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