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
internally and

Answer

To find the position vector of point R that divides the line segment joining points P and Q in the ratio 2:1 internally, we can use the section formula. The section formula states that if a point R divides the line segment joining points P and Q in the ratio m:n, then the position vector of R can be calculated using the formula:

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

### Step-by-Step Solution:

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

2. **Assign the values of m and n:**
- Here, \( m = 2 \) and \( n = 1 \) (since R divides PQ in the ratio 2:1).

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

4. **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:
\[ 2\vec{a} + 2\vec{b} + 3\vec{a} - 2\vec{b} = (2\vec{a} + 3\vec{a}) + (2\vec{b} - 2\vec{b}) = 5\vec{a} \]
- Now, substitute back into the formula:
\[ \vec{R} = \frac{5\vec{a}}{3} \]

5. **Final Result:**
\[ \vec{R} = \frac{5}{3}\vec{a} \]