Find the position vector of the point, which divides the join of points with position vectors `3vec(a)-2vec(b)` and `2vec(a)+3vec(b)` in the ratio 2 : 1.

Answer

To find the position vector of the point that divides the line segment joining the points with position vectors \( \vec{P_1} = 3\vec{a} - 2\vec{b} \) and \( \vec{P_2} = 2\vec{a} + 3\vec{b} \) in the ratio \( 2:1 \), we can use the section formula in vector algebra.

### Step-by-Step Solution:

1. **Identify the position vectors**:
- Let \( \vec{P_1} = 3\vec{a} - 2\vec{b} \)
- Let \( \vec{P_2} = 2\vec{a} + 3\vec{b} \)

2. **Assign the ratio**:
- The ratio in which the point divides the line segment is \( m_1 : m_2 = 2 : 1 \).
- Here, \( m_1 = 2 \) and \( m_2 = 1 \).

3. **Apply the section formula**:
The position vector \( \vec{R} \) of the point dividing the line segment joining \( \vec{P_1} \) and \( \vec{P_2} \) in the ratio \( m_1 : m_2 \) is given by:
\[ \vec{R} = \frac{m_2 \vec{P_1} + m_1 \vec{P_2}}{m_1 + m_2} \]

4. **Substitute the values**:
\[ \vec{R} = \frac{1(3\vec{a} - 2\vec{b}) + 2(2\vec{a} + 3\vec{b})}{2 + 1} \]

5. **Calculate the numerator**:
- First, calculate \( 1(3\vec{a} - 2\vec{b}) = 3\vec{a} - 2\vec{b} \)
- Next, calculate \( 2(2\vec{a} + 3\vec{b}) = 4\vec{a} + 6\vec{b} \)
- Now, combine these:
\[ 3\vec{a} - 2\vec{b} + 4\vec{a} + 6\vec{b} = (3\vec{a} + 4\vec{a}) + (-2\vec{b} + 6\vec{b}) = 7\vec{a} + 4\vec{b} \]

6. **Calculate the denominator**:
\[ m_1 + m_2 = 2 + 1 = 3 \]

7. **Final expression for \( \vec{R} \)**:
\[ \vec{R} = \frac{7\vec{a} + 4\vec{b}}{3} \]

### Conclusion:
The position vector of the point that divides the line segment joining the points with position vectors \( 3\vec{a} - 2\vec{b} \) and \( 2\vec{a} + 3\vec{b} \) in the ratio \( 2:1 \) is:
\[ \vec{R} = \frac{7\vec{a} + 4\vec{b}}{3} \]