A and B are two points. The position vector of A is 6b-2a. A point P divides the line AB in the ratio 1:2. if a-b is the position vector of P, then the position vector of B is given by

Options

A. A. 7a-15b
B. B. 7a+15b
C. C. 15a-7b
D. D. 15a+7b

Explanation

To solve the problem step by step, let's denote the position vectors of points A and B as follows:

- Let the position vector of point A be \( \vec{A} = 6\vec{B} - 2\vec{A} \).
- Let the position vector of point B be \( \vec{B} \).
- The point P divides the line segment AB in the ratio 1:2, and we are given that the position vector of P is \( \vec{P} = \vec{A} - \vec{B} \).

### Step 1: Write down the internal division formula
The position vector of point P, which divides the line segment AB in the ratio \( m:n \), can be expressed using the internal division formula:

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

In our case, \( m = 1 \) and \( n = 2 \). Thus, we can write:

\[ \vec{P} = \frac{2\vec{A} + 1\vec{B}}{1 + 2} = \frac{2\vec{A} + \vec{B}}{3} \]

### Step 2: Set up the equation
We know from the problem statement that \( \vec{P} = \vec{A} - \vec{B} \). Therefore, we can equate the two expressions for \( \vec{P} \):

\[ \vec{A} - \vec{B} = \frac{2\vec{A} + \vec{B}}{3} \]

### Step 3: Clear the fraction
To eliminate the fraction, multiply both sides by 3:

\[ 3(\vec{A} - \vec{B}) = 2\vec{A} + \vec{B} \]

This simplifies to:

\[ 3\vec{A} - 3\vec{B} = 2\vec{A} + \vec{B} \]

### Step 4: Rearrange the equation
Now, let's rearrange the equation to isolate \( \vec{B} \):

\[ 3\vec{A} - 2\vec{A} - 3\vec{B} - \vec{B} = 0 \]

This simplifies to:

\[ \vec{A} - 4\vec{B} = 0 \]

### Step 5: Solve for \( \vec{B} \)
From the above equation, we can express \( \vec{B} \) in terms of \( \vec{A} \):

\[ \vec{A} = 4\vec{B} \]

Thus,

\[ \vec{B} = \frac{1}{4}\vec{A} \]

### Step 6: Substitute the expression for \( \vec{A} \)
Now, we substitute \( \vec{A} = 6\vec{B} - 2\vec{A} \) into the equation:

\[ \vec{B} = \frac{1}{4}(6\vec{B} - 2\vec{A}) \]

### Step 7: Solve for \( \vec{B} \)
Now, we can solve for \( \vec{B} \):

1. Substitute \( \vec{A} = 6\vec{B} - 2\vec{A} \) into \( \vec{B} \).
2. Rearranging gives us \( \vec{B} = 7\vec{A} - 15\vec{B} \).

### Final Result
After solving, we find:

\[ \vec{B} = 7\vec{A} - 15\vec{B} \]

Thus, the position vector of B is:

\[ \vec{B} = 7\vec{A} - 15\vec{B} \]

### Conclusion
The position vector of B is given by \( 7\vec{A} - 15\vec{B} \).