The position vectors of the points A,B, and C are `hati+2hatj-hatk, hati+hatj+hatk`, and `2hati+3hatj+2hatk` respectively. If A is chosen as the origin, then the position vectors B and C are

Options

A. `veci+2hatk, hati+hatj+3hatk`
B. `hatj+2hatk, hati+hatj+3hatk`
C. `-hatj+2hatk, hati-hatj+3hatk`
D. `-hatj+2hatk,hati+hatj+3hatk`

Explanation

To find the position vectors of points B and C with respect to point A, we will follow these steps:

1. **Identify the position vectors of points A, B, and C**:
- The position vector of point A is given as \( \vec{OA} = \hat{i} + 2\hat{j} - \hat{k} \).
- The position vector of point B is given as \( \vec{OB} = \hat{i} + \hat{j} + \hat{k} \).
- The position vector of point C is given as \( \vec{OC} = 2\hat{i} + 3\hat{j} + 2\hat{k} \).

2. **Calculate the position vector of B with respect to A**:
- The position vector \( \vec{AB} \) can be calculated using the formula:
\[ \vec{AB} = \vec{OB} - \vec{OA} \]
- Substituting the values:
\[ \vec{AB} = (\hat{i} + \hat{j} + \hat{k}) - (\hat{i} + 2\hat{j} - \hat{k}) \]
- Simplifying this:
\[ \vec{AB} = \hat{i} + \hat{j} + \hat{k} - \hat{i} - 2\hat{j} + \hat{k} \]
\[ \vec{AB} = 0\hat{i} - \hat{j} + 2\hat{k} \]
- Thus, the position vector of B with respect to A is:
\[ \vec{AB} = -\hat{j} + 2\hat{k} \]

3. **Calculate the position vector of C with respect to A**:
- The position vector \( \vec{AC} \) can be calculated using the formula:
\[ \vec{AC} = \vec{OC} - \vec{OA} \]
- Substituting the values:
\[ \vec{AC} = (2\hat{i} + 3\hat{j} + 2\hat{k}) - (\hat{i} + 2\hat{j} - \hat{k}) \]
- Simplifying this:
\[ \vec{AC} = 2\hat{i} + 3\hat{j} + 2\hat{k} - \hat{i} - 2\hat{j} + \hat{k} \]
\[ \vec{AC} = (2\hat{i} - \hat{i}) + (3\hat{j} - 2\hat{j}) + (2\hat{k} + \hat{k}) \]
\[ \vec{AC} = \hat{i} + \hat{j} + 3\hat{k} \]
- Thus, the position vector of C with respect to A is:
\[ \vec{AC} = \hat{i} + \hat{j} + 3\hat{k} \]

4. **Final Result**:
- The position vectors of points B and C with respect to point A are:
\[ \vec{AB} = -\hat{j} + 2\hat{k} \]
\[ \vec{AC} = \hat{i} + \hat{j} + 3\hat{k} \]