If the position vector of the poinot A is a+2b and a point P with position vector `vec(a)` divides a line segement AB in the ratio `2:3` then the position vector of the point B is

Options

A. `2 vec(a) - vec(b)`
B. `vec( b) - 2 vec(a)`
C. `vec(a) - 3 vec(b)`
D. `vec(b)`

Answer

To find the position vector of point B given the position vector of point A and the position vector of point P that divides the line segment AB in the ratio 2:3, we can follow these steps:

### Step 1: Define the position vectors
Let the position vector of point A be given as:
\[ \vec{A} = \vec{a} + 2\vec{b} \]
Let the position vector of point B be denoted as:
\[ \vec{B} = \vec{x} \]
The position vector of point P is given as:
\[ \vec{P} = \vec{a} \]

### Step 2: Use the section formula
Since point P divides the line segment AB in the ratio 2:3, we can use the section formula. The formula states that if a point divides a line segment joining points A and B in the ratio m:n, then the position vector of the point is given by:
\[ \vec{P} = \frac{n\vec{A} + m\vec{B}}{m+n} \]
In our case, m = 2 and n = 3. Therefore, we have:
\[ \vec{a} = \frac{3(\vec{a} + 2\vec{b}) + 2\vec{x}}{2 + 3} \]

### Step 3: Simplify the equation
Substituting the values into the equation, we get:
\[ \vec{a} = \frac{3(\vec{a} + 2\vec{b}) + 2\vec{x}}{5} \]
Multiplying both sides by 5 to eliminate the denominator:
\[ 5\vec{a} = 3(\vec{a} + 2\vec{b}) + 2\vec{x} \]

### Step 4: Expand and rearrange
Expanding the right side:
\[ 5\vec{a} = 3\vec{a} + 6\vec{b} + 2\vec{x} \]
Now, rearranging the equation to isolate \(2\vec{x}\):
\[ 2\vec{x} = 5\vec{a} - 3\vec{a} - 6\vec{b} \]
This simplifies to:
\[ 2\vec{x} = 2\vec{a} - 6\vec{b} \]

### Step 5: Solve for \(\vec{x}\)
Dividing both sides by 2 gives:
\[ \vec{x} = \vec{a} - 3\vec{b} \]

### Conclusion
Thus, the position vector of point B is:
\[ \vec{B} = \vec{a} - 3\vec{b} \]