The position vector of the point which divides the join of points with position vectors `bara+barb` and `2 bara - barb` in the ratio `1:2` is

Options

A. `(3 bara+2barb)/(3)`
B. `bara `
C. `(5bara-barb)/(3)`
D. `(4 bara+barb)/(3)`

Answer

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

### Step-by-step Solution:

1. **Identify the Position Vectors:**
- Let \( \mathbf{A_1} = \mathbf{a} + \mathbf{b} \)
- Let \( \mathbf{A_2} = 2\mathbf{a} - \mathbf{b} \)

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

3. **Apply the Section Formula:**
The position vector \( \mathbf{P} \) of the point dividing the line segment joining \( \mathbf{A_1} \) and \( \mathbf{A_2} \) in the ratio \( m_1:m_2 \) is given by:
\[ \mathbf{P} = \frac{m_2 \mathbf{A_1} + m_1 \mathbf{A_2}}{m_1 + m_2} \]
Substituting the values:
\[ \mathbf{P} = \frac{2(\mathbf{a} + \mathbf{b}) + 1(2\mathbf{a} - \mathbf{b})}{1 + 2} \]

4. **Simplify the Expression:**
- Calculate the numerator:
\[ \mathbf{P} = \frac{2\mathbf{a} + 2\mathbf{b} + 2\mathbf{a} - \mathbf{b}}{3} \]
- Combine like terms:
\[ \mathbf{P} = \frac{(2\mathbf{a} + 2\mathbf{a}) + (2\mathbf{b} - \mathbf{b})}{3} = \frac{4\mathbf{a} + \mathbf{b}}{3} \]

5. **Final Answer:**
The position vector of the point that divides the line segment in the ratio \( 1:2 \) is:
\[ \mathbf{P} = \frac{4\mathbf{a} + \mathbf{b}}{3} \]