If the position vector of a point A is `vec a + 2 vec b and vec a ` divides AB in the ratio `2:3`, then the position vector of B, is

Options

A. ` vec a - vec b`
B. `vec b - 2 vec a `
C. `vec a - 3 vec b `
D. ` vec b`

Answer

To solve the problem, we need to find the position vector of point B given that point A has a position vector of \( \vec{a} + 2\vec{b} \) and that point A divides the line segment AB in the ratio \( 2:3 \).

### Step-by-step Solution:

1. **Identify the position vector of point A**:
The position vector of point A is given as:
\[ \vec{A} = \vec{a} + 2\vec{b} \]

2. **Use the section formula**:
Since point A divides the line segment AB in the ratio \( 2:3 \), we can use the section formula to find the position vector of point B. The section formula states that if a point divides the line segment joining two points in the ratio \( m:n \), then the position vector of the dividing point is given by:
\[ \vec{P} = \frac{n\vec{A} + m\vec{B}}{m+n} \]
In our case, we have \( m = 3 \) (for B) and \( n = 2 \) (for A).

3. **Set up the equation**:
Let the position vector of point B be \( \vec{B} \). According to the section formula, we can express the position vector of point A as:
\[ \vec{A} = \frac{3\vec{B} + 2(\vec{a} + 2\vec{b})}{2 + 3} \]
Simplifying the denominator gives us:
\[ \vec{A} = \frac{3\vec{B} + 2\vec{a} + 4\vec{b}}{5} \]

4. **Multiply through by 5**:
To eliminate the fraction, we multiply both sides by 5:
\[ 5\vec{A} = 3\vec{B} + 2\vec{a} + 4\vec{b} \]

5. **Substitute for \(\vec{A}\)**:
Substitute \(\vec{A} = \vec{a} + 2\vec{b}\) into the equation:
\[ 5(\vec{a} + 2\vec{b}) = 3\vec{B} + 2\vec{a} + 4\vec{b} \]

6. **Expand and rearrange**:
Expanding the left side gives:
\[ 5\vec{a} + 10\vec{b} = 3\vec{B} + 2\vec{a} + 4\vec{b} \]
Rearranging the equation to isolate \( \vec{B} \):
\[ 3\vec{B} = 5\vec{a} + 10\vec{b} - 2\vec{a} - 4\vec{b} \]
Simplifying the right side:
\[ 3\vec{B} = (5\vec{a} - 2\vec{a}) + (10\vec{b} - 4\vec{b}) = 3\vec{a} + 6\vec{b} \]

7. **Divide by 3**:
Finally, divide both sides by 3 to solve for \( \vec{B} \):
\[ \vec{B} = \vec{a} + 2\vec{b} \]

### Conclusion:
The position vector of point B is:
\[ \vec{B} = \vec{a} + 2\vec{b} \]