1. **State the problem:** We are given quadrilateral OACB with vectors \( \vec{OA} = 4a \), \( \vec{OB} = 3b \), and \( \vec{BC} = 2a + b \). We need to find the vector \( \vec{AC} \) in terms of \( a \) and \( b \) in simplest form. 2. **Understand vector positions:** \( \vec{OC} \) can be found because point C connects B and C starting from O: \[ \vec{OC} = \vec{OB} + \vec{BC} = 3b + (2a + b) = 2a + 4b \] 3. **Calculate \( \vec{AC} \):** By definition, \[ \vec{AC} = \vec{OC} - \vec{OA} \] Substitute the known vectors: \[ \vec{AC} = (2a + 4b) - 4a = 2a + 4b - 4a = -2a + 4b \] 4. **Simplify the expression:** \[ \vec{AC} = -2a + 4b \] This is the vector \( \vec{AC} \) expressed in terms of \( a \) and \( b \). **Final answer:** \[ \boxed{\vec{AC} = -2a + 4b} \]