If \( \vec{a}, \vec{b}\) and \(\vec{c} \) are three vectors such that \( \vec{a} + \vec{b} + \vec{c} = \vec{0} \), \( |\vec{a}| = 7 \), \( |\vec{b}| = 3 \) and \( |\vec{c}| = 5 \), then angle between \( \vec{b} \) and \( \vec{c} \) is

\( \frac{\pi}{3} \)

The correct answer is Option (4) → \( \frac{\pi}{3} \)

Given:

$\vec{a} + \vec{b} + \vec{c} = \vec{0} \Rightarrow \vec{a} = -(\vec{b} + \vec{c})$

Taking modulus on both sides:

$|\vec{a}|^2 = |\vec{b} + \vec{c}|^2$

Using identity: $|\vec{b} + \vec{c}|^2 = |\vec{b}|^2 + |\vec{c}|^2 + 2|\vec{b}||\vec{c}|\cos\theta$

Substitute values:

$7^2 = 3^2 + 5^2 + 2(3)(5)\cos\theta$

$49 = 9 + 25 + 30\cos\theta$

$49 = 34 + 30\cos\theta$

$15 = 30\cos\theta$

$\cos\theta = \frac{1}{2}$

$\theta = 60^\circ$