Let two non-collinear unit vectors - and b- form an acute angle- A point P moves so that at any time t the position vector OP where- O is the origin is given by - cos t -b-sin t- When P is farthest from origin O- let M be the length of OP and - be the unit vector along OP- Then-A- -b-b- and M -1-b-1-2B- -b-b- and M -1-b-1-2C- -b-b- and M -1-2 - - b-1-2D- -b- b - and M -1-2 -b-1-2

The correct option is B û = â+b̂/|â+b̂| and M = (1+ â.b̂)^1/2 Given, O⃗P⃗= â cos t + b̂ sin t ⇒ |O⃗P⃗|=√((â.â) cos^2 t + b̂.b̂ sin^2 t +2 â.b̂ sin t cos t) ...

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Standard IX

Mathematics

Dot Product of Two Vectors

Let two non-c...

Question

Let two non-collinear unit vectors $^a and^b$ form an acute angle. A point P moves so that at any time t the position vector
$\to O P$ (where, O is the origin) is given by
$^a cos t +^b sin t$ . When P is farthest from origin O, let M be the length of $\to O P$ and $^u$ be the unit vector along $\to O P$ . Then,

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

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

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

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

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Solution

The correct option is A
$^u = \frac{^a +^b}{|^a +^b |} and M = (1 +^a .^b)^{\frac{1}{2}}$

Given, $\to O P =^a cos t +^b sin t \Rightarrow | \to O P | = \sqrt{(^a .^a) {cos}^{2} t +^b .^b {sin}^{2} t + 2^a .^b sin t cos t} \Rightarrow | \to O P | = \sqrt{1 +^a .^b sin 2 t} \Rightarrow | \to O P |_{max} = M = \sqrt{1 +^a .^b} at sin 2 t = 1 \Rightarrow t = \frac{π}{4} At t = \frac{π}{4}, \to O P = \frac{1}{\sqrt{2}} (^a +^b)$
Unit vector along $\to O P$ at $(t = \frac{π}{4}) = \frac{^a +^b}{|^a +^b |}$

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Let two non-collinear unit vectors ^a and ^b form an acute angle. A point P moves so that at any time t the position vector →OP (where, O is the origin) is given by ^acos t+^bsin t. When P is farthest from origin O, let M be the length of →OP and ^u be the unit vector along →OP. Then,

The correct option is A ^u=^a+^b|^a+^b| and M=(1+^a.^b)12 Given, →OP=^a cos t+^b sin t⇒|→OP|=√(^a.^a) cos2 t+^b.^b sin2 t+2^a.^b sin t cos t⇒|→OP|=√1+^a.^b sin 2t⇒|→OP|max=M=√1+^a.^b at sin 2t=1⇒t=π4At t=π4, →OP=1√2(^a+^b) Unit vector along →OP at (t=π4)=^a+^b|^a+^b|