If a and b are position vectors of A and B respectively the position vector of a point C on AB produced such that `vec(AC)=3 vec(AB)` is

Options

A. `3vecb-2veca`
B. `2veca-3vecb`
C. `2veca+3vecb`
D. `3veca-2vecb`

Answer

To find the position vector of point C on the line segment AB produced such that \( \vec{AC} = 3 \vec{AB} \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Vectors**:
- Let \( \vec{a} \) be the position vector of point A.
- Let \( \vec{b} \) be the position vector of point B.
- We need to find the position vector \( \vec{c} \) of point C.

2. **Express the Vectors**:
- The vector \( \vec{AB} \) can be expressed as:
\[ \vec{AB} = \vec{b} - \vec{a} \]

3. **Use the Given Condition**:
- According to the problem, we have:
\[ \vec{AC} = 3 \vec{AB} \]
- Substituting the expression for \( \vec{AB} \):
\[ \vec{AC} = 3(\vec{b} - \vec{a}) \]

4. **Express \( \vec{AC} \) in Terms of \( \vec{c} \)**:
- The vector \( \vec{AC} \) can also be expressed as:
\[ \vec{AC} = \vec{c} - \vec{a} \]
- Therefore, we have:
\[ \vec{c} - \vec{a} = 3(\vec{b} - \vec{a}) \]

5. **Rearranging the Equation**:
- Rearranging gives:
\[ \vec{c} - \vec{a} = 3\vec{b} - 3\vec{a} \]
- Adding \( \vec{a} \) to both sides:
\[ \vec{c} = 3\vec{b} - 3\vec{a} + \vec{a} \]
- Simplifying this:
\[ \vec{c} = 3\vec{b} - 2\vec{a} \]

6. **Final Result**:
- Thus, the position vector of point C is:
\[ \vec{c} = 3\vec{b} - 2\vec{a} \]