The position vector of the point which divides the join of points `2veca-3vecbandveca+vecb` in the ratio 3:1, is

Options

A. `(3veca-2vecb)/(2)`
B. `(7veca-8vecb)/(7)`
C. `(3veca)/(4)`
D. `(5veca)/(4)`

Explanation

To find the position vector of the point that divides the line segment joining the points \( \vec{P} = 2\vec{a} - 3\vec{b} \) and \( \vec{Q} = \vec{a} + \vec{b} \) in the ratio \( 3:1 \), we can use the section formula.

### Step-by-Step Solution:

1. **Identify the Points**:
- Let \( \vec{P} = 2\vec{a} - 3\vec{b} \)
- Let \( \vec{Q} = \vec{a} + \vec{b} \)

2. **Use the Section Formula**:
The position vector \( \vec{R} \) of the point that divides the line segment joining \( \vec{P} \) and \( \vec{Q} \) in the ratio \( m:n \) is given by:
\[ \vec{R} = \frac{m\vec{Q} + n\vec{P}}{m+n} \]
Here, \( m = 3 \) and \( n = 1 \).

3. **Substitute the Values**:
Substitute \( \vec{P} \) and \( \vec{Q} \) into the formula:
\[ \vec{R} = \frac{3(\vec{a} + \vec{b}) + 1(2\vec{a} - 3\vec{b})}{3 + 1} \]

4. **Simplify the Expression**:
- Calculate \( 3(\vec{a} + \vec{b}) \):
\[ 3\vec{a} + 3\vec{b} \]
- Calculate \( 1(2\vec{a} - 3\vec{b}) \):
\[ 2\vec{a} - 3\vec{b} \]
- Combine these results:
\[ \vec{R} = \frac{(3\vec{a} + 3\vec{b}) + (2\vec{a} - 3\vec{b})}{4} \]
- Combine like terms:
\[ \vec{R} = \frac{(3\vec{a} + 2\vec{a}) + (3\vec{b} - 3\vec{b})}{4} = \frac{5\vec{a}}{4} \]

5. **Final Result**:
The position vector of the point \( R \) is:
\[ \vec{R} = \frac{5\vec{a}}{4} \]