The position vectors of A and B are `hati-hatj+2hatk and 3hati-hatj+3hatk`. The position vector of the middle points of the line AB is

Options

A. `(1)/(2)hati-(1)/(2)hatj+hatk`
B. `2hati-hatj+(5)/(2)hatk`
C. `(3)/(2)hati-(1)/(2)hatj+(3)/(2)hatk`
D. none of these

Explanation

To find the position vector of the midpoint of the line segment AB, we can follow these steps:

### Step 1: Identify the position vectors of points A and B
The position vector of point A is given as:
\[ \vec{OA} = \hat{i} - \hat{j} + 2\hat{k} \]
The position vector of point B is given as:
\[ \vec{OB} = 3\hat{i} - \hat{j} + 3\hat{k} \]

### Step 2: Use the midpoint formula
The position vector of the midpoint C of the line segment AB can be calculated using the formula:
\[ \vec{OC} = \frac{\vec{OA} + \vec{OB}}{2} \]

### Step 3: Substitute the position vectors into the formula
Now, we substitute the position vectors of A and B into the midpoint formula:
\[ \vec{OC} = \frac{(\hat{i} - \hat{j} + 2\hat{k}) + (3\hat{i} - \hat{j} + 3\hat{k})}{2} \]

### Step 4: Combine the vectors
Combine the components of the vectors:
\[ \vec{OC} = \frac{(\hat{i} + 3\hat{i}) + (-\hat{j} - \hat{j}) + (2\hat{k} + 3\hat{k})}{2} \]
This simplifies to:
\[ \vec{OC} = \frac{(4\hat{i}) + (-2\hat{j}) + (5\hat{k})}{2} \]

### Step 5: Divide each component by 2
Now, divide each component by 2:
\[ \vec{OC} = \frac{4}{2}\hat{i} + \frac{-2}{2}\hat{j} + \frac{5}{2}\hat{k} \]
This simplifies to:
\[ \vec{OC} = 2\hat{i} - \hat{j} + \frac{5}{2}\hat{k} \]

### Final Answer
Thus, the position vector of the midpoint C of the line segment AB is:
\[ \vec{OC} = 2\hat{i} - \hat{j} + \frac{5}{2}\hat{k} \]