If the position vectors of the points A and B are `hati+3hatj-hatk and 3hati-hatj-3hatk`, then what will be the position vector of the mid-point of AB

Options

A. `hati+2hatj-hatk`
B. `2hati+hatj-2hatk`
C. `2hati+hatj-hatk`
D. `hati+hatj-2hatk`

Explanation

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

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

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

3. **Substitute the position vectors into the formula.**
- Substitute the values of \(\vec{OA}\) and \(\vec{OB}\):
\[ \vec{OC} = \frac{(\hat{i} + 3\hat{j} - \hat{k}) + (3\hat{i} - \hat{j} - 3\hat{k})}{2} \]

4. **Combine the vectors.**
- Combine the components of the vectors:
\[ \vec{OC} = \frac{(1 + 3)\hat{i} + (3 - 1)\hat{j} + (-1 - 3)\hat{k}}{2} \]
- This simplifies to:
\[ \vec{OC} = \frac{4\hat{i} + 2\hat{j} - 4\hat{k}}{2} \]

5. **Divide each component by 2.**
- Now divide each component by 2:
\[ \vec{OC} = 2\hat{i} + 1\hat{j} - 2\hat{k} \]

6. **Write the final answer.**
- Therefore, the position vector of the midpoint C is:
\[ \vec{OC} = 2\hat{i} + \hat{j} - 2\hat{k} \]

### Final Answer:
The position vector of the midpoint of AB is \(2\hat{i} + \hat{j} - 2\hat{k}\).