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Let $\vec{A} = (\hat{i}+\hat{j}) and , \vec{B}= (2\hat{i}- \hat{j})$ The magnitude of a coplanar vector $\vec{C}$

such that $\vec{A}\cdot \vec{C}=\vec{B}\cdot \vec{C}=\vec{A}\cdot \vec{B}$ is given by

Option 1)
$\sqrt(10/9)$

Option 2)
$\sqrt(5/9)$

Option 3)
$\sqrt(20/9)$

Option 4)
$\sqrt(9/12)$

Answers (1)

As we have learned

Scalar , Dot or Inner Product -

Scalar product of two vector $\vec{A}$ & $\vec{B}$ written as $\vec{A}$ $\cdot$ $\vec{B}$ is a scalar quantity given by the product of magnitude of $\vec{A}$ & $\vec{B}$ and the cosine of smaller angle between them.

$\vec{A}\cdot \vec{B}= A\, B\cdot \cos \Theta$

- wherein

showing representation of scalar products of vectors.

Let $\vec{c}= x\hat{i}+y\hat{j}$

$\vec{A}\cdot \vec{C}= x+y$

$\vec{B}\cdot \vec{C}= 2x-y$

$\vec{A}\cdot \vec{B}= 2-1=1$

$\Rightarrow x+y=1$

$2x-y=1$

---------------------------

$3x=2$ or x = 2/3

y= 1/3

$\therefore \vec{c}= 2/3\hat{i}+1/3\hat{j}$
$\therefore| \vec{c}|=\sqrt{\left ( 4/9+1/9 \right )}=\sqrt{\left ( 5/9 \right )}$

Option 1)

$\sqrt(10/9)$

This is incorrect

Option 2)

$\sqrt(5/9)$

This is correct

Option 3)

$\sqrt(20/9)$

This is incorrect

Option 4)

$\sqrt(9/12)$

This is incorrect

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Vakul

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