Question

Let two non-collinear unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ form an acute angle. A point P moves, so that at any time t the positive 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 $M=(1+\hat{a}\cdot \hat{b}{)}^{1/2}$
• B. $\hat{u}=\frac{\hat{a}-\hat{b}}{|\hat{a}-\hat{b}|}$ and $M=(1+\hat{a}\cdot \hat{b}{)}^{1/2}$
• C. $\hat{u}=\frac{\hat{a}+\hat{b}}{|\hat{a}+\hat{b}|}$ and $M=(1+2\hat{a}\cdot \hat{b}{)}^{1/2}$
• D. $\hat{u}=\frac{\hat{a}-\hat{b}}{|\hat{a}-\hat{b}|}$ and $M=(1+2\hat{a}\cdot \hat{b}{)}^{1/2}$

Answer 1

Answer: (A)

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

$\left|\overline{OP}\right|=\left|\hat{a}\cos t+\hat{b}\sin t\right|$

$={\left({\cos }^{2}t+{\sin }^{2}t+2\cos t\sin t\hat{a}\cdot \hat{b}\right)}^{1/2}$

$={\left(1+2\cos t\sin t\hat{a}\cdot \hat{b}\right)}^{1/2}$

$={\left(1+\sin 2t\hat{a}\cdot \hat{b}\right)}^{1/2}$

${\left|\overline{OP}\right|}_{max}={\left(1+\hat{a}\cdot \hat{b}\right)}^{1/2}when,t=\frac{\pi }{4}$

$\hat{u}=\frac{\left(\hat{a}+\hat{b}\right)}{\sqrt{2}\frac{\left|\hat{a}+\hat{b}\right|}{\sqrt{2}}}$

$\Rightarrow \hat{u}=\frac{\left(\hat{a}+\hat{b}\right)}{\left|\hat{a}+\hat{b}\right|}$