Question

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

• A. 6
• B. 10
• C. 7
• D. 8

Answer 1

The Correct Option is D

Step 1. Area of parallelogram \( S \) with adjacent sides \( OA \) and \( OC \):
    \(S = |\vec{a} \times \vec{b}|\)
Step 2. Area of quadrilateral \( OABC \):

  \(\text{Area of } OABC = \text{Area of } \triangle OAB + \text{Area of } \triangle OBC\)
\(= \frac{1}{2} \left| \vec{a} \times (12\vec{a} + 4\vec{b}) \right| + \frac{1}{2} \left| \vec{b} \times (12\vec{a} + 4\vec{b}) \right|\)
\(= \frac{1}{2} |4\vec{a} \times \vec{b}| + \frac{1}{2} |12\vec{a} \times \vec{b}|\)
\(= 8|\vec{a} \times \vec{b}|\)

Step 3. Ratio:

\(\text{Ratio} = \frac{\text{Area of quadrilateral } OABC}{\text{Area of parallelogram } S} = \frac{8|\vec{a} \times \vec{b}|}{|\vec{a} \times \vec{b}|} = 8\)

The Correct Answer is: 8