Let `vec(a) and vec(b)` be the position vectors of A and B respectively. If C is the point`3 vec(a) - 2vec(b)`, then which one of the following is correct?

Options

A. A. C is in between A and B
B. B. A is in between C and B
C. C. B is in between A and C
D. D. A, B and C are not collinear

Answer

To solve the problem, we need to analyze the position vectors of points A, B, and C, where C is defined as \( C = 3\vec{a} - 2\vec{b} \).

### Step-by-Step Solution:

1. **Define Position Vectors**:
Let \( \vec{A} \) and \( \vec{B} \) be the position vectors of points A and B, respectively. We denote the position vector of point C as \( \vec{C} \).

2. **Express C in terms of A and B**:
According to the problem, we have:
\[ \vec{C} = 3\vec{A} - 2\vec{B} \]

3. **Rearranging the Equation**:
We can rearrange the equation to express it in a different form:
\[ \vec{C} + 2\vec{B} = 3\vec{A} \]
This shows a relationship between the vectors.

4. **Divide by 3**:
To isolate \( \vec{A} \), we divide the entire equation by 3:
\[ \vec{A} = \frac{\vec{C} + 2\vec{B}}{3} \]

5. **Interpret the Result**:
The equation \( \vec{A} = \frac{\vec{C} + 2\vec{B}}{3} \) indicates that the position vector \( \vec{A} \) is a weighted average of \( \vec{C} \) and \( \vec{B} \). Specifically, it suggests that point A is located one-third of the way from point C to point B.

6. **Conclusion**:
Since \( \vec{A} \) is between \( \vec{C} \) and \( \vec{B} \), we conclude that point A lies between points C and B.

### Final Answer:
The correct option is that A is between C and B.

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