A and B are two points with position vectors `2vec(a)-3vec(b)` and `6vec(b)-vec(a)` respectively. Write the position vector of a point, which divides the line segment AB internally in the ratio 1 : 2.

Answer

To find the position vector of a point that divides the line segment AB internally in the ratio 1:2, we will follow these steps:

### Step 1: Identify the position vectors of points A and B
Given:
- Position vector of point A: \( \vec{A} = 2\vec{a} - 3\vec{b} \)
- Position vector of point B: \( \vec{B} = 6\vec{b} - \vec{a} \)

### Step 2: Use the section formula
The section formula states that if a point divides the line segment joining two points \( A \) and \( B \) in the ratio \( m:n \), then the position vector \( \vec{R} \) of the point is given by:

\[ \vec{R} = \frac{m\vec{B} + n\vec{A}}{m+n} \]

In this case, we have \( m = 1 \) and \( n = 2 \).

### Step 3: Substitute the values into the formula
Substituting the values of \( \vec{A} \) and \( \vec{B} \) into the section formula:

\[ \vec{R} = \frac{1(6\vec{b} - \vec{a}) + 2(2\vec{a} - 3\vec{b})}{1 + 2} \]

### Step 4: Simplify the expression
Now, we simplify the numerator:

1. Calculate \( 1(6\vec{b} - \vec{a}) \):
\[ = 6\vec{b} - \vec{a} \]

2. Calculate \( 2(2\vec{a} - 3\vec{b}) \):
\[ = 4\vec{a} - 6\vec{b} \]

3. Combine both results:
\[ \vec{R} = \frac{(6\vec{b} - \vec{a}) + (4\vec{a} - 6\vec{b})}{3} \]

4. Combine like terms:
\[ = \frac{(6\vec{b} - 6\vec{b}) + (-\vec{a} + 4\vec{a})}{3} \]
\[ = \frac{3\vec{a}}{3} \]
\[ = \vec{a} \]

### Final Answer
The position vector of the point that divides the line segment AB in the ratio 1:2 is \( \vec{R} = \vec{a} \).

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