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

To solve the problem, we need to find the areas of the parallelogram \(S\) and the quadrilateral \(OABC\), then compare their ratios. 

1. Begin by identifying vectors. From the question: \(\overrightarrow{OA} = \vec{a}\), \(\overrightarrow{OB} = 12\vec{a} + 4\vec{b}\), and \(\overrightarrow{OC} = \vec{b}\).
2. The area of parallelogram \(S\) with sides \( \overrightarrow{OA} \) and \( \overrightarrow{OC} \) is given by the magnitude of the cross product: \(\text{Area of } S = |\vec{a} \times \vec{b}|\).
3. For quadrilateral \(OABC\), split it into two triangles: \( \Delta OAB \) and \( \Delta OBC \).
   1. Area of \(\Delta OAB = \frac{1}{2} |\overrightarrow{OA} \times \overrightarrow{OB}|\). \(\overrightarrow{OB} = 12\vec{a} + 4\vec{b}\). Therefore: \(\overrightarrow{OA} \times \overrightarrow{OB} = \vec{a} \times (12\vec{a} + 4\vec{b}) = 4(\vec{a} \times \vec{b})\). So, the area of \(\Delta OAB = 2|\vec{a} \times \vec{b}|\).
   2. Area of \(\Delta OBC = \frac{1}{2} |\overrightarrow{OB} \times \overrightarrow{OC}|\). Calculate \(\overrightarrow{OB} \times \overrightarrow{OC} = (12\vec{a} + 4\vec{b}) \times \vec{b} = 12 (\vec{a} \times \vec{b})\), because \(\vec{b} \times \vec{b} = 0\). Thus, the area of \(\Delta OBC = 6|\vec{a} \times \vec{b}|\).
4. So, the area of quadrilateral \(OABC\) is the sum of the areas of the two triangles: \(2|\vec{a} \times \vec{b}| + 6|\vec{a} \times \vec{b}| = 8|\vec{a} \times \vec{b}|\).
5. Finally, the ratio of the area of quadrilateral \(OABC\) to parallelogram \(S\) is: \(\frac{8|\vec{a} \times \vec{b}|}{|\vec{a} \times \vec{b}|} = 8\).

Thus, the correct answer is 8.

Answer 2

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