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  4. If | veca+ vecb|2 + | veca ⋅ vecb|2 =144 | veca| =6 then | vecb| is equal to

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Q. If ${\left|\vec{a}+\vec{b}\right|^2}$ + ${\left|\vec{a} \cdot \vec{b}\right|^2} =144$ $|\vec{a}| =6 $ then $|\vec{b}|$ is equal to

KCETKCET 2020

Solution:

$|\vec{a} \times \vec{b}|^{2}+|\vec{a} \cdot \vec{b}|^{2}=144$
$\Rightarrow |\vec{a}|^{2}|\vec{b}|^{2} \sin ^{2} \theta+|\vec{a}|^{2}|\vec{b}|^{2} \cos ^{2} \theta=144$
$\Rightarrow |\vec{a}|^{2}|\vec{b}|^{2}=144$
$\Rightarrow |\vec{b}|^{2}=\frac{144}{36}=4 \,\,\,(\because|\vec{a}|=6)$
$\Rightarrow |\vec{b}|=2$