Step-by-step Solution:

To find the magnitude of \(\vec{a} - \vec{b}\) given that \(|\vec{a}| = |\vec{b}| = |\vec{c}| = 2\) and \(\vec{c} = \vec{a} + \vec{b}\), we can proceed with the following steps: 1. Given Information: \[ |\vec{a}| = |\vec{b}| = |\vec{c}| = 2 \] \[ \vec{c} = \vec{a} + \vec{b} \] 2. Use the Law of Cosines: The magnitude of \(\vec{c}\) can be expressed using the law of cosines: \[ |\vec{c}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + 2|\vec{a}||\vec{b}|\cos\theta \] where \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\). Substituting the given magnitudes: \[ 2^2 = 2^2 + 2^2 + 2 \cdot 2 \cdot 2 \cos\theta \] \[ 4 = 4 + 4 + 8\cos\theta \] \[ 4 = 8 + 8\cos\theta \] \[ 8\cos\theta = -4 \] \[ \cos\theta = -\frac{1}{2} \] 3. Find the Magnitude of \(\vec{a} - \vec{b}\): The magnitude of \(\vec{a} - \vec{b}\) can also be found using the law of cosines: \[ |\vec{a} - \vec{b}|^2 = |\vec{a}|^2 + |\vec{b}|^2 - 2|\vec{a}||\vec{b}|\cos\theta \] Substituting the known values: \[ |\vec{a} - \vec{b}|^2 = 2^2 + 2^2 - 2 \cdot 2 \cdot 2 \cdot \left(-\frac{1}{2}\right) \] \[ |\vec{a} - \vec{b}|^2 = 4 + 4 + 4 = 12 \] \[ |\vec{a} - \vec{b}| = \sqrt{12} = 2\sqrt{3} \] Therefore, the magnitude of \(\vec{a} - \vec{b}\) is: \[ \boxed{2\sqrt{3}} \]