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

Explanation

To solve the problem, we need to analyze the position vectors of points A, B, and C. Given that the position vectors of points A and B are represented by \(\vec{a}\) and \(\vec{b}\) respectively, and point C is defined as:

\[ \vec{c} = 3\vec{a} - 2\vec{b} \]

We need to determine the relationship between points A, B, and C.

### Step-by-step Solution:

1. **Define the Position Vectors**:
- Let \(\vec{A} = \vec{a}\) (position vector of point A)
- Let \(\vec{B} = \vec{b}\) (position vector of point B)
- Let \(\vec{C} = 3\vec{a} - 2\vec{b}\) (position vector of point C)

2. **Find the Vector \(\vec{AB}\)**:
- The vector from A to B is given by:
\[ \vec{AB} = \vec{B} - \vec{A} = \vec{b} - \vec{a} \]

3. **Find the Vector \(\vec{AC}\)**:
- The vector from A to C is given by:
\[ \vec{AC} = \vec{C} - \vec{A} = (3\vec{a} - 2\vec{b}) - \vec{a} = 2\vec{a} - 2\vec{b} \]
- Simplifying gives:
\[ \vec{AC} = 2(\vec{a} - \vec{b}) \]

4. **Relate \(\vec{AC}\) and \(\vec{AB}\)**:
- We can express \(\vec{AC}\) in terms of \(\vec{AB}\):
\[ \vec{AC} = 2(\vec{A} - \vec{B}) = -2(\vec{B} - \vec{A}) = -2\vec{AB} \]

5. **Interpret the Result**:
- The expression \(\vec{AC} = -2\vec{AB}\) indicates that the vector \(\vec{AC}\) is in the opposite direction to \(\vec{AB}\) and is twice as long.
- This means that point C lies outside the segment AB, specifically on the extension of the line through A and B.

6. **Conclusion**:
- Since C is outside the segment AB, we can conclude that point A lies between points B and C.

### Final Answer:
Thus, the correct statement is:
- A lies between C and B.