SATHEE: Vectors 1 Question 9

Vectors 1 Question 9

10. 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 $\vec{O P}$ (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 $\vec{O P}$ and $\hat{u}$ be the unit vector along $\vec{O P}$ . Then,

(2008, 3M)

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

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

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Answer:

Correct Answer: 10. (a)

Solution:

Given, $\vec{O P} = \hat{a} \cos t + \hat{b} \sin t$

$\Rightarrow | \vec{O P} | = \sqrt{(\hat{a} \cdot \hat{a}) \cos^{2} t + (\hat{b} \cdot \hat{b}) \sin^{2} t + 2 \hat{a} \cdot \hat{b} \sin t \cos t}$

$\Rightarrow | \vec{O P} | = \sqrt{1 + \hat{a} \cdot \hat{b} \sin 2 t}$

$\Rightarrow | \vec{O P} |_{max} = M = \sqrt{1 + \hat{a} \cdot \hat{b}}$ at $\sin 2 t = 1 \Rightarrow t = \frac{π}{4}$

At $t = \frac{π}{4}, \vec{O P} = \frac{1}{\sqrt{2}} (\hat{a} + \hat{b})$

Unit vector along $\vec{O P}$ at $(t = \frac{π}{4}) = \frac{\hat{a} + \hat{b}}{| \hat{a} + \hat{b} |}$

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