Let two non-collinear unit vector $\hat{a}$ a n d $\hat{b}$ form an acute angle. A point $P$ moves so that at any time $t,$ the position vector $OP(whereO$ is the origin$)$ is given by $\hat{a}\cos t+\hat{b}\sin t\dot{W}henP$ is farthest from origin $O,$ let $M$ be the length of $OPand\hat{u}$ be the unit vector along $OP\dot{}$ Then (a)$\hat{u}=\frac{\hat{a}+\hat{b}}{\left|\hat{a}+\hat{b}\right|}andM={\left(1+\hat{a}\dot{\hat{b}}\right)}^{1/2}$ (b) \displaystyle\hat{{u}}=\frac{{\hat{{a}}-\hat{{b}}}}{{{\left|\hat{{a}}-\hat{{b}}\right|}}}{a}{n}{d}{M}={\left({1}+\hat{{a}}\dot{{\hat}}\right)}^{{{1}/{2}}}(c) $\hat{u}=\frac{\hat{a}+\hat{b}}{\left|\hat{a}+\hat{b}\right|}andM={\left(1+2\hat{a}\dot{\hat{b}}\right)}^{1/2}$(d) $\hat{u}=\frac{\hat{a}-\hat{b}}{\left|\hat{a}-\hat{b}\right|}andM={\left(1+2\hat{a}\dot{\hat{b}}\right)}^{1/2}$

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