Step 1: Understanding the Question:
We need to relate the area of a general quadrilateral $ABCD$ to the area of a parallelogram defined by its sides $\vec{a}$ and $\vec{b}$.
Step 2: Key Formula or Approach:
Area of parallelogram with adjacent sides $\vec{a}, \vec{b}$ is $|\vec{a} \times \vec{b}|$.
Area of quadrilateral $ABCD = \frac{1}{2} |\vec{AC} \times \vec{BD}|$.
Step 3: Detailed Explanation:
1. We are given $\vec{AB} = \vec{a}$, $\vec{AD} = \vec{b}$, and $\vec{AC} = 2\vec{a} + 3\vec{b}$.
2. Find vector $\vec{BD}$:
$\vec{BD} = \vec{AD} - \vec{AB} = \vec{b} - \vec{a}$.
3. Calculate the area of quadrilateral $ABCD$:
$Area_{quad} = \frac{1}{2} |(2\vec{a} + 3\vec{b}) \times (\vec{b} - \vec{a})|$
$Area_{quad} = \frac{1}{2} |2(\vec{a} \times \vec{b}) - 2(\vec{a} \times \vec{a}) + 3(\vec{b} \times \vec{b}) - 3(\vec{b} \times \vec{a})|$
Since $\vec{v} \times \vec{v} = \vec{0}$ and $\vec{b} \times \vec{a} = -(\vec{a} \times \vec{b})$:
$Area_{quad} = \frac{1}{2} |2(\vec{a} \times \vec{b}) - 0 + 0 + 3(\vec{a} \times \vec{b})| = \frac{1}{2} |5(\vec{a} \times \vec{b})| = \frac{5}{2} |\vec{a} \times \vec{b}|$.
4. We are given $Area_{quad} = \alpha \times Area_{para}$, and $Area_{para} = |\vec{a} \times \vec{b}|$.
Comparing terms: $\alpha = \frac{5}{2}$.
Step 4: Final Answer:
The value of $\alpha$ is $\frac{5}{2}$.