Let O be an interior point of `DeltaABC` such that `bar(OA)+2bar(OB) + 3bar(OC) = 0`. Then the ratio of a `DeltaABC` to area of `DeltaAOC` is

Options

A. 2
B. `3/2`
C. 3
D. `5/2`

Explanation

To solve the problem, we need to find the ratio of the area of triangle ABC to the area of triangle AOC given that \( \vec{OA} + 2\vec{OB} + 3\vec{OC} = 0 \).

### Step-by-Step Solution:

1. **Understanding the Given Equation**:
We have the equation \( \vec{OA} + 2\vec{OB} + 3\vec{OC} = 0 \). This implies that we can express \( \vec{OA} \) in terms of \( \vec{OB} \) and \( \vec{OC} \):
\[ \vec{OA} = -2\vec{OB} - 3\vec{OC} \]

2. **Setting Up the Area of Triangle ABC**:
The area of triangle ABC can be expressed using the cross product:
\[ \text{Area}_{ABC} = \frac{1}{2} |\vec{AB} \times \vec{AC}| \]
where \( \vec{AB} = \vec{B} - \vec{A} \) and \( \vec{AC} = \vec{C} - \vec{A} \).

3. **Expressing Vectors**:
Substitute \( \vec{A} = \vec{OA} \), \( \vec{B} = \vec{OB} \), and \( \vec{C} = \vec{OC} \):
\[ \vec{AB} = \vec{B} - \vec{A} = \vec{OB} - \vec{OA} \]
\[ \vec{AC} = \vec{C} - \vec{A} = \vec{OC} - \vec{OA} \]

4. **Calculating Area of Triangle AOC**:
The area of triangle AOC can also be expressed using the cross product:
\[ \text{Area}_{AOC} = \frac{1}{2} |\vec{OA} \times \vec{OC}| \]

5. **Substituting for \( \vec{OA} \)**:
Substitute \( \vec{OA} = -2\vec{OB} - 3\vec{OC} \) into the area formula for triangle AOC:
\[ \text{Area}_{AOC} = \frac{1}{2} |(-2\vec{OB} - 3\vec{OC}) \times \vec{OC}| \]

6. **Using the Properties of Cross Product**:
The cross product \( \vec{OC} \times \vec{OC} = 0 \), thus:
\[ \text{Area}_{AOC} = \frac{1}{2} |(-2\vec{OB}) \times \vec{OC}| \]
\[ = \frac{1}{2} \cdot 2 |\vec{OB} \times \vec{OC}| = |\vec{OB} \times \vec{OC}| \]

7. **Finding the Ratio of Areas**:
Now, we can find the ratio of the areas:
\[ \frac{\text{Area}_{ABC}}{\text{Area}_{AOC}} = \frac{\frac{1}{2} |\vec{AB} \times \vec{AC}|}{|\vec{OB} \times \vec{OC}|} \]

8. **Using the Relationship**:
From the equation \( \vec{OA} + 2\vec{OB} + 3\vec{OC} = 0 \), we can deduce that the coefficients (1, 2, 3) represent the relative areas of the triangles formed. Thus, the ratio of the area of triangle ABC to the area of triangle AOC is:
\[ \frac{\text{Area}_{ABC}}{\text{Area}_{AOC}} = 3 \]

### Final Answer:
The ratio of the area of triangle ABC to the area of triangle AOC is \( 3 \).