If $\vec{a}, \vec{b}$ are non-zero vectors of same magnitude such that the angle between $\vec{a} and \vec{b} is \frac{2 π}{3} and \vec{a} . \vec{b} = - 8, then |\vec{a}| =______________.$

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Solution

Given:
$|\vec{a}| = |\vec{b}|$ ...(1)
The angle between the vectors $\vec{a} and \vec{b} is \frac{2 π}{3}$ ...(2)

$\vec{a} \cdot \vec{b} = - 8$ ....(3)

$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos θ \Rightarrow - 8 = {|\vec{a}|}^{2} \cos \frac{2 π}{3} (From (1), (2) and (3)) \Rightarrow - 8 = {|\vec{a}|}^{2} (- \frac{1}{2}) \Rightarrow {|\vec{a}|}^{2} = 16 \Rightarrow |\vec{a}| = 4$

Hence, $|\vec{a}| = 4$ .

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