A, B have vectors ` vec a , vec b ` relative to the origin O and X, Y divide ` vec(AB)` internally and externally respectively in the ratio `2:1` . Then , `vec (XY)=`

Options

A. `(3)/(2) (vec b - vec a) `
B. `(4)/(3) (vec a - vec b) `
C. `(5)/(6) (vec b - vec a) `
D. `(4)/(3) (vec b - vec a) `

Answer

To solve the problem, we need to find the vector \(\vec{XY}\) where points \(X\) and \(Y\) divide the line segment \(\vec{AB}\) internally and externally in the ratio \(2:1\).

### Step-by-Step Solution

1. **Identify the Position Vectors**:
Let the position vector of point \(A\) be \(\vec{a}\) and the position vector of point \(B\) be \(\vec{b}\). The origin \(O\) is represented by the vector \(\vec{0}\).

2. **Find the Position Vector of Point \(X\)** (Internal Division):
Since \(X\) divides \(\vec{AB}\) internally in the ratio \(2:1\), we can use the section formula:
\[ \vec{x} = \frac{m\vec{b} + n\vec{a}}{m+n} \]
where \(m = 2\) and \(n = 1\).
\[ \vec{x} = \frac{2\vec{b} + 1\vec{a}}{2 + 1} = \frac{2\vec{b} + \vec{a}}{3} \]

3. **Find the Position Vector of Point \(Y\)** (External Division):
Since \(Y\) divides \(\vec{AB}\) externally in the ratio \(2:1\), we can use the external division formula:
\[ \vec{y} = \frac{m\vec{b} - n\vec{a}}{m-n} \]
where \(m = 2\) and \(n = 1\).
\[ \vec{y} = \frac{2\vec{b} - 1\vec{a}}{2 - 1} = 2\vec{b} - \vec{a} \]

4. **Calculate the Vector \(\vec{XY}\)**:
The vector \(\vec{XY}\) is given by:
\[ \vec{XY} = \vec{y} - \vec{x} \]
Substituting the values of \(\vec{y}\) and \(\vec{x}\):
\[ \vec{XY} = \left(2\vec{b} - \vec{a}\right) - \left(\frac{2\vec{b} + \vec{a}}{3}\right) \]

5. **Simplify the Expression**:
To simplify, we need a common denominator:
\[ \vec{XY} = \left(2\vec{b} - \vec{a}\right) - \left(\frac{2\vec{b}}{3} + \frac{\vec{a}}{3}\right) \]
\[ = \left(2\vec{b} - \frac{2\vec{b}}{3}\right) - \left(\vec{a} + \frac{\vec{a}}{3}\right) \]
\[ = \left(\frac{6\vec{b}}{3} - \frac{2\vec{b}}{3}\right) - \left(\frac{3\vec{a}}{3} + \frac{\vec{a}}{3}\right) \]
\[ = \frac{4\vec{b}}{3} - \frac{4\vec{a}}{3} \]
\[ = \frac{4}{3}(\vec{b} - \vec{a}) \]

### Final Answer:
Thus, the vector \(\vec{XY}\) is:
\[ \vec{XY} = \frac{4}{3}(\vec{b} - \vec{a}) \]