Find the position of R, which divides the line joining `P(3vec(a)-2vec(b))` and `Q(vec(a)+vec(b))` in the ratio 2 : 1
externally.

Answer

To find the position vector of point R, which divides the line segment joining points P and Q in the ratio 2:1 externally, we will use the section formula for external division.

### Step-by-Step Solution:

1. **Identify the position vectors of points P and Q**:
- Let \( \vec{P} = 3\vec{a} - 2\vec{b} \)
- Let \( \vec{Q} = \vec{a} + \vec{b} \)

2. **Understand the ratio of division**:
- The point R divides the line segment PQ in the ratio 2:1 externally. Here, \( m = 2 \) and \( n = 1 \).

3. **Apply the formula for external division**:
- The position vector of point R, which divides the line segment PQ externally in the ratio \( m:n \), is given by:
\[ \vec{R} = \frac{m\vec{Q} - n\vec{P}}{m - n} \]

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

5. **Simplify the expression**:
- Calculate the numerator:
\[ 2(\vec{a} + \vec{b}) = 2\vec{a} + 2\vec{b} \]
\[ -1(3\vec{a} - 2\vec{b}) = -3\vec{a} + 2\vec{b} \]
- Combine these:
\[ 2\vec{a} + 2\vec{b} - 3\vec{a} + 2\vec{b} = (2\vec{a} - 3\vec{a}) + (2\vec{b} + 2\vec{b}) = -\vec{a} + 4\vec{b} \]

6. **Final calculation**:
- Now, divide by \( 2 - 1 = 1 \):
\[ \vec{R} = -\vec{a} + 4\vec{b} \]

### Final Answer:
The position vector of point R is:
\[ \vec{R} = -\vec{a} + 4\vec{b} \]