If `vec(a) and vec(b)` are position vectors of the points A and B respectively, then what is the position vector of a point C on AB produced such that `vec(AC) = vec(2AB)`?

Options

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

Explanation

To find the position vector of point C on the line segment AB produced such that \( \vec{AC} = 2\vec{AB} \), we can follow these steps:

### Step 1: Define the position vectors
Let:
- \( \vec{A} \) be the position vector of point A.
- \( \vec{B} \) be the position vector of point B.

### Step 2: Find the vector \( \vec{AB} \)
The vector \( \vec{AB} \) can be expressed as:
\[ \vec{AB} = \vec{B} - \vec{A} \]

### Step 3: Express the vector \( \vec{AC} \)
Since we know that \( \vec{AC} = 2\vec{AB} \), we can substitute \( \vec{AB} \) into this equation:
\[ \vec{AC} = 2(\vec{B} - \vec{A}) \]

### Step 4: Relate \( \vec{AC} \) to \( \vec{C} \)
Using the definition of the position vector, we can express \( \vec{AC} \) in terms of \( \vec{C} \):
\[ \vec{AC} = \vec{C} - \vec{A} \]

### Step 5: Set the two expressions for \( \vec{AC} \) equal
Now we can set the two expressions for \( \vec{AC} \) equal to each other:
\[ \vec{C} - \vec{A} = 2(\vec{B} - \vec{A}) \]

### Step 6: Rearrange to solve for \( \vec{C} \)
Rearranging the equation gives:
\[ \vec{C} = 2(\vec{B} - \vec{A}) + \vec{A} \]
\[ \vec{C} = 2\vec{B} - 2\vec{A} + \vec{A} \]
\[ \vec{C} = 2\vec{B} - \vec{A} \]

### Final Answer
Thus, the position vector of point C is:
\[ \vec{C} = 2\vec{B} - \vec{A} \]