# Let be two unit vectors such that if    then       is equal to: Option 1) Option 2) Option 3) Option 4)

As we learnt in

Vector Product of two vectors(cross product) -

If $\vec{a}$ and $\vec{b}$ are two vectors and $\Theta$ is the angle between them , then $\vec{a}\times \vec{b}=\left |\vec{a} \left | \right |\vec{b} \right |Sin\Theta \hat{n}$

- wherein

$\hat{n}$ is unit vector perpendicular to both $\vec{a} \: and \: \vec{b}$

and

Scalar Product of two vectors (dot product) -

$\vec{a}\vec{b}=\left | a \right |\left | b \right |Cos\theta$

- wherein

$\Theta$ is the angle between the vectors$\vec{a}\: and\:\vec{b}$

$\left | \vec{a}+\vec{b} \right |=\sqrt{3}$

Squaring both sides

$\left | \vec{a} \right |^{2}+\left | \vec{b} \right |^{2}+2\left | \vec{a} \right |\left | \vec{b} \right |cos \theta =3$

$cos \theta=\frac{1}{2}$    $\Rightarrow \theta=\frac{\pi }{3}$

$\vec{c}=\vec{a}+2\vec{b}+3\left | \vec{a} \right |\left | \vec{b} \right | sin \frac{\pi }{3}\hat{n}$

$\vec{c}=\vec{a}+2\vec{b}+\frac{3\sqrt{3}}{2}\hat{n}$

$\vec{c}\cdot \vec{c}=\left ( \vec{a}+2\vec{b}+\frac{3\sqrt{3}}{2}\hat{n} \right )\cdot \left (\vec{a}+2\vec{b}+\frac{3\sqrt{3}}{2}\hat{n} \right )$

$\left | \vec{c} \right |^{2}=\left | \vec{a} \right |^{2}+4\left | \vec{b} \right |^{2}+\frac{27}{4}+4\left ( \vec{a}\cdot \vec{b} \right )$

$\left | \vec{c} \right |^{2}=1+4+\frac{27}{4}+4\times 1\times 1\times \frac{1}{2}$

$\left | \vec{c} \right |^{2}=7+\frac{27}{4}=\frac{55}{4}$

$\left | \vec{c} \right |^{2}=\frac{\sqrt{55}}{2}$

$2\left | \vec{c} \right |=\sqrt{55}$

Option 1)

This option is correct.

Option 2)

This option is incorrect.

Option 3)

This option is incorrect.

Option 4)

This option is incorrect.

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