1. The problem asks to find the vector expression for $AD$ given that $AB = \vec{a}$ and $CD = 2AB = 2\vec{a}$. We need to express $AD$ in terms of $\vec{a}$ and $\vec{b}$. 2. Recall that in a quadrilateral $ABCD$, the vector $AD$ can be expressed as the sum of vectors $AB$ and $BD$: $$\vec{AD} = \vec{AB} + \vec{BD}$$ 3. Since $BD$ is the vector from $B$ to $D$, and $BC = \vec{b}$, we can write $BD$ as: $$\vec{BD} = \vec{BC} + \vec{CD} = \vec{b} + 2\vec{a}$$ 4. Substitute $\vec{AB} = \vec{a}$ and $\vec{BD} = \vec{b} + 2\vec{a}$ into the expression for $\vec{AD}$: $$\vec{AD} = \vec{a} + (\vec{b} + 2\vec{a}) = 3\vec{a} + \vec{b}$$ 5. Therefore, the vector $AD$ is $3\vec{a} + \vec{b}$. Final answer: C. $3a + b$