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

Answer 1

Answer: (A)

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

$OP$ is maximum when $M$ is maximum,
$M=\sqrt{co{s}^{2}t+si{n}^{2}t+2costsint(a.b)}=\sqrt{1+2costsint(a.b)}$
This is maximum when $costsint$ is maximum, its maximum value is $\frac{1}{2}$
Hence, $M=\sqrt{1+a.b}$
$\hat{u}=\hat{a}cost+\hat{b}sint=\frac{1}{\sqrt{2}}(\hat{a}+\hat{b})$
Unit vector is $\frac{\frac{1}{\sqrt{2}}(\hat{a}+\hat{b})}{\frac{1}{\sqrt{2}}|a+b|}=\frac{\hat{a}+\hat{b}}{|a+b|}$