P and Q are two points with positon vectors `3 veca - 2 vecb and veca + vecb` respectively. Write the positon vector of a point R which divides the segment PQ in the ratio `2:1` extermally.

Answer

To find the position vector of point R which divides the segment PQ in the ratio 2:1 externally, we can use the section formula for external division. The formula for the position vector of a point R that divides the line segment joining points P and Q externally in the ratio m:n is given by:

\[ \vec{R} = \frac{m\vec{Q} - n\vec{P}}{m - n} \]

### Step-by-Step Solution:

1. **Identify the Position Vectors:**
- The position vector of point P is given as \( \vec{P} = 3\vec{a} - 2\vec{b} \).
- The position vector of point Q is given as \( \vec{Q} = \vec{a} + \vec{b} \).

2. **Set the Ratio:**
- We need to divide the segment PQ in the ratio \( 2:1 \) externally. Here, \( m = 2 \) and \( n = 1 \).

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

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

5. **Final Result:**
- Thus, the position vector of point R is:
\[ \vec{R} = -\vec{a} + 4\vec{b} \]