Let `vec (OA)= vec a`, `vec (OB) = 12 vec a + 4 vec b` and `vec (OC) = vec b`, where O is the origin. If S is the parallelogram with adjacent sides OA and OC, then `frac"{ area of quadrilateral OABC}""{area of S}"` is equal to ___

Options

A. 6
B. 10
C. 7
D. 8

Answer

To solve the problem, we need to find the ratio of the area of quadrilateral OABC to the area of the parallelogram S formed by vectors OA and OC.

### Step-by-Step Solution:

1. **Define Vectors**:
- Let \(\vec{OA} = \vec{a}\)
- Let \(\vec{OB} = 12\vec{a} + 4\vec{b}\)
- Let \(\vec{OC} = \vec{b}\)

2. **Identify Points**:
- Point O is the origin (0,0).
- Point A corresponds to the vector \(\vec{a}\).
- Point B corresponds to the vector \(12\vec{a} + 4\vec{b}\).
- Point C corresponds to the vector \(\vec{b}\).

3. **Find Vectors for Quadrilateral OABC**:
- The vector \(\vec{AB} = \vec{OB} - \vec{OA} = (12\vec{a} + 4\vec{b}) - \vec{a} = 11\vec{a} + 4\vec{b}\)
- The vector \(\vec{CB} = \vec{OB} - \vec{OC} = (12\vec{a} + 4\vec{b}) - \vec{b} = 12\vec{a} + 3\vec{b}\)

4. **Calculate the Area of Quadrilateral OABC**:
- The area of a quadrilateral can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \left| \vec{d_1} \times \vec{d_2} \right| \]
where \(\vec{d_1}\) and \(\vec{d_2}\) are the diagonals.
- The diagonals are:
- \(\vec{AC} = \vec{OC} - \vec{OA} = \vec{b} - \vec{a}\)
- \(\vec{OB} = 12\vec{a} + 4\vec{b}\)
- Thus, the area of quadrilateral OABC is:
\[ \text{Area}_{OABC} = \frac{1}{2} \left| (12\vec{a} + 4\vec{b}) \times (\vec{b} - \vec{a}) \right| \]

5. **Calculate the Cross Product**:
- We compute:
\[ (12\vec{a} + 4\vec{b}) \times (\vec{b} - \vec{a}) = 12\vec{a} \times \vec{b} - 12\vec{a} \times \vec{a} + 4\vec{b} \times \vec{b} - 4\vec{b} \times \vec{a} \]
- Since \(\vec{a} \times \vec{a} = 0\) and \(\vec{b} \times \vec{b} = 0\), we simplify to:
\[ = 12\vec{a} \times \vec{b} - 4\vec{b} \times \vec{a} = 12\vec{a} \times \vec{b} + 4\vec{a} \times \vec{b} = 16\vec{a} \times \vec{b} \]
- Therefore,
\[ \text{Area}_{OABC} = \frac{1}{2} \left| 16\vec{a} \times \vec{b} \right| = 8 \left| \vec{a} \times \vec{b} \right| \]

6. **Calculate the Area of Parallelogram S**:
- The area of parallelogram S formed by vectors \(\vec{OA}\) and \(\vec{OC}\) is given by:
\[ \text{Area}_S = \left| \vec{a} \times \vec{b} \right| \]

7. **Find the Ratio**:
- Now, we find the ratio of the areas:
\[ \frac{\text{Area}_{OABC}}{\text{Area}_S} = \frac{8 \left| \vec{a} \times \vec{b} \right|}{\left| \vec{a} \times \vec{b} \right|} = 8 \]

### Final Answer:
The ratio \(\frac{\text{Area of quadrilateral OABC}}{\text{Area of S}} = 8\).

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