If $\vec a+\vec b+\vec c=\vec 0$ and $|\vec a| = 3, |\vec b| = 5, |\vec c| = 7$, then the angle between $\vec a$ and $\vec b$ is

$\frac{\pi}{3}$

The correct answer is Option (2) → $\frac{\pi}{3}$

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

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

$7^2 = 3^2 + 5^2 + 2(\vec a\cdot \vec b)$

$49 = 9 + 25 + 2(\vec a\cdot \vec b)$

$49 = 34 + 2(\vec a\cdot \vec b)$

$2(\vec a\cdot \vec b)=15$

$\vec a\cdot \vec b=\frac{15}{2}$

$\vec a\cdot \vec b = |\vec a||\vec b|\cos\theta$

$\frac{15}{2}=3\cdot 5\cos\theta$

$\frac{15}{2}=15\cos\theta$

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

$\theta = \frac{\pi}{3}$