Step-by-step Solution:

Given that \( \vec a + \vec b + \vec c \) is collinear with \( \vec d \), we have: \[ \vec a + \vec b + \vec c = \lambda \vec d \quad \text{(1)} \] Also, \( \vec b + \vec c + \vec d \) is collinear with \( \vec a \), so: \[ \vec b + \vec c + \vec d = \mu \vec a \quad \text{(2)} \] From equation (1) minus equation (2): \[ \vec a - \vec d = \lambda \vec d - \mu \vec a \] By comparison, we get \( \lambda = -1 \) and \( \mu = -1 \). Substitute \( \lambda = -1 \) into equation (1): \[ \vec a + \vec b + \vec c = - \vec d \] Thus: \[ \Rightarrow \vec a + \vec b + \vec c + \vec d = 0 \]