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 of B and C are

Options

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

Explanation

To solve the problem, we need to find the position vectors of points B and C when point A is chosen as the origin. The position vectors of points A, B, and C are given as follows:

- Position vector of A: \( \vec{A} = \hat{i} + 2\hat{j} - \hat{k} \)
- Position vector of B: \( \vec{B} = \hat{i} + \hat{j} + \hat{k} \)
- Position vector of C: \( \vec{C} = 2\hat{i} + 3\hat{j} + 2\hat{k} \)

### Step-by-Step Solution:

1. **Set A as the Origin**:
When A is chosen as the origin, we can denote the position vector of A as \( \vec{O} = 0\hat{i} + 0\hat{j} + 0\hat{k} \).

2. **Find the Position Vector of B**:
The position vector of B relative to A (now the origin) can be calculated using the formula:
\[ \vec{OB} = \vec{B} - \vec{A} \]
Substituting the values:
\[ \vec{OB} = (\hat{i} + \hat{j} + \hat{k}) - (\hat{i} + 2\hat{j} - \hat{k}) \]
Simplifying this:
\[ \vec{OB} = \hat{i} + \hat{j} + \hat{k} - \hat{i} - 2\hat{j} + \hat{k} \]
\[ = (1 - 1)\hat{i} + (1 - 2)\hat{j} + (1 + 1)\hat{k} \]
\[ = 0\hat{i} - 1\hat{j} + 2\hat{k} \]
Thus, the position vector of B relative to A is:
\[ \vec{OB} = -\hat{j} + 2\hat{k} \]

3. **Find the Position Vector of C**:
Similarly, we find the position vector of C relative to A:
\[ \vec{OC} = \vec{C} - \vec{A} \]
Substituting the values:
\[ \vec{OC} = (2\hat{i} + 3\hat{j} + 2\hat{k}) - (\hat{i} + 2\hat{j} - \hat{k}) \]
Simplifying this:
\[ \vec{OC} = 2\hat{i} + 3\hat{j} + 2\hat{k} - \hat{i} - 2\hat{j} + \hat{k} \]
\[ = (2 - 1)\hat{i} + (3 - 2)\hat{j} + (2 + 1)\hat{k} \]
\[ = 1\hat{i} + 1\hat{j} + 3\hat{k} \]
Thus, the position vector of C relative to A is:
\[ \vec{OC} = \hat{i} + \hat{j} + 3\hat{k} \]

### Final Answers:
- Position vector of B relative to A: \( \vec{OB} = -\hat{j} + 2\hat{k} \)
- Position vector of C relative to A: \( \vec{OC} = \hat{i} + \hat{j} + 3\hat{k} \)